Solve Quadratics by Multiple Methods - Algebra 2

1. The equation $x^2 - 8x + k = 0$ is solved by completing the square. If the resulting form is $(x - 4)^2 = 9$, what is the value of $k$, and what method would be most efficient for solving the original equation?
   Answer: $k = 7$; factoring is most efficient since $x^2 - 8x + 7 = (x - 1)(x - 7)$ When you encounter a quadratic equation and are given information about completing the square, you need to work backwards to find the missing coefficient, then determine the most efficient solving method. Starting with the completed square form $(x - 4)^2 = 9$, let's expand this back to standard form. Expanding gives us $x^2 - 8x + 16 = 9$, which rearranges to $x^2 - 8x + 7 = 0$. Therefore, $k = 7$. Now for the solving method: when $k = 7$, we have $x^2 - 8x + 7 = 0$. Notice that we need two numbers that multiply to 7 and add to -8. Those numbers are -1 and -7, so this factors neatly as $(x - 1)(x - 7) = 0$. Since factoring is typically the fastest method when it works cleanly, this makes factoring the most efficient approach. Choice A correctly identifies $k = 7$ but incorrectly suggests the quadratic formula is most efficient. While the discriminant being a perfect square does make the quadratic formula work nicely, factoring is still faster when the factorization is simple. Choice C gets $k$ wrong by confusing the constant term in the completed square form (16) with the value of $k$ in the original equation. Choice D incorrectly calculates $k = -7$ and suggests completing the square is most efficient. While the even coefficient does make completing the square straightforward, factoring remains the fastest method. Study tip: When a quadratic factors easily with integer roots, factoring beats all other methods for speed, even if other approaches work well too.
2. When solving $x^2 + 6x - 1 = 0$ by completing the square, a student writes $x^2 + 6x + 9 = 1 + 9$, then $(x + 3)^2 = 10$. What should be the student's next step to find the exact solutions?
   Answer: Take the square root: $x + 3 = \pm\sqrt{10}$, so $x = -3 \pm \sqrt{10}$ When taking the square root of both sides of (x + 3)² = 10, we must include both positive and negative square roots: x + 3 = ±√10. This gives x = -3 ± √10, yielding both solutions. Choice A only gives one solution by forgetting the ± symbol. Choice C goes backward in the process. Choice D is unnecessarily complicated and doesn't directly solve for x.
3. Solve by taking square roots: $(x-3)^2 = 5$.​
   Answer: $x=3\pm\sqrt{5}$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For (x-3) squared = 5, take square roots on both sides: x-3 = ± sqrt(5), so x = 3 ± sqrt(5)—remember the ± for both solutions. This method is perfect here since it's already in squared form. Choice A correctly applies the square root method, including both positive and negative roots. Choice D fails by omitting the positive root, giving only one solution—quadratics usually have two solutions, so always include ±! Remember, when taking square roots, if the right side is positive, you get two real solutions; if zero, one; if negative, complex. Practice by rewriting equations into squared form when possible for efficiency.
4. Solve $x^2 - 7x + 10 = 0$ by factoring or by the quadratic formula. Which pair of solutions is correct?​
   Answer: $x=2$ and $x=5$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For x squared - 7x + 10 = 0, factors to (x-2)(x-5)=0 (2*5=10, 2+5=7), so x=2 and x=5; formula confirms [7 ± sqrt(49-40)]/2 = [7 ± 3]/2 = 5 or 2. Both methods work here. Choice B correctly gives the solutions from either method. Choice A uses wrong signs for factors, leading to negative roots—test by adding, not subtracting! Versatility tip: try factoring first if easy, then formula to verify. This builds confidence in answers.
5. The equation $2x^2 - 8x + m = 0$ has solutions that are complex numbers with imaginary parts of $\pm\frac{\sqrt{6}}{2}$. What is the value of $m$?
   Answer: $m = 11$ Using the quadratic formula: x = (8 ± √(64 - 8m))/4 = (8 ± √(64 - 8m))/4 = 2 ± √(64 - 8m)/4. For complex solutions with imaginary part ±√6/2, we need 64 - 8m < 0 and √(8m - 64)/4 = √6/2. Solving: √(8m - 64) = 2√6, so 8m - 64 = 24, giving 8m = 88, so m = 11. Choice A gives imaginary part ±√11/2. Choice B gives imaginary part ±√7/2. Choice C gives imaginary part ±√12/4 = ±√3/2.
6. Solve by factoring: $x^2 - 9x + 20 = 0$.
   Answer: $x = 4, 5$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving x squared - 9x + 20 = 0 by factoring: find two numbers that multiply to 20 and add to -9. Since both must be negative (positive product, negative sum), try: -1 and -20 (add to -21, no), -2 and -10 (add to -12, no), -4 and -5 (add to -9, yes!). So x squared - 9x + 20 = (x - 4)(x - 5) = 0, giving x = 4 or x = 5. Choice A correctly identifies x = 4, 5 as the solutions by factoring (x - 4)(x - 5) = 0. Choice B incorrectly has negative solutions x = -4, -5, which would come from factoring x squared + 9x + 20 = (x + 4)(x + 5) = 0, not our equation. Remember: when factoring x squared + bx + c, if b is negative and c is positive, both factors have the form (x - positive number). Quick factoring check: does 4 times 5 = 20? Yes! Does -4 + (-5) = -9? Yes! So (x - 4)(x - 5) = x squared - 9x + 20. The sign pattern in the factored form matches the signs in the expanded form.
7. Solve by completing the square: $x^2+6x+1=0.$
   Answer: $x=-3\pm 2\sqrt{2}$ This question targets completing the square, a method that rewrites quadratics as $(x + h)^2 = k$, then takes square roots—useful for deriving the formula or when factoring fails. For $x^2 + 6x + 1 = 0$, move 1: $x^2 + 6x = -1$, add $(\frac{6}{2})^2 = 9$: $(x+3)^2 = 8$, $x+3 = \pm \sqrt{8} = \pm 2\sqrt{2}$, $x = -3 \pm 2\sqrt{2}$—perfect! This method shines for monic quadratics without even coefficients. Choice C correctly simplifies $\sqrt{8}$ to $2\sqrt{2}$. Choice A forgets to simplify the radical—always reduce for clean answers. Practice by halving b and squaring to complete efficiently. You've got the hang of it—keep building!
8. Solve the quadratic equation by factoring: $x^2 - 9x + 20 = 0$
   Answer: $x=4$ or $x=5$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's $x^2 =$ number, inspect or take square roots ($x = \pm \sqrt{\text{number}}$), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like $x^2 + 5x + 6$), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving $x^2 - 9x + 20 = 0$ by factoring: look for two numbers that multiply to 20 and add to -9, which are -4 and -5, so $(x-4)(x-5)=0$, thus $x=4$ or $x=5$. Choice B correctly identifies the factored form and applies the zero product property to find both positive real roots. Choice A misses one root by perhaps only considering one factor or forgetting the or statement—remember, quadratics usually have two solutions unless discriminant is zero! Transferable strategy: always check if the constant and coefficient suggest easy factors (products to c, sum to b), and verify by expanding back to original. You're building efficiency—great job practicing factoring for when it fits perfectly!
9. Use the quadratic formula to solve: $2x^2-3x-5=0$
   Answer: $x=\dfrac{3\pm\sqrt{49}}{4}$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's $x^2 = \text{number}$, inspect or take square roots ($x = \pm \sqrt{\text{number}}$), (2) if it's $(\text{expression})^2 = \text{number}$, take square roots, (3) if it factors easily (like $x^2 + 5x + 6$), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For $2x^2 - 3x - 5 = 0$ using quadratic formula: $a=2$, $b=-3$, $c=-5$, discriminant $9 + 40=49$, $x=\frac{3 \pm 7}{4}$ (since $-b=3$). Choice A correctly plugs in with $-b$ positive and denominator 4. Choice B uses $+b$ instead of $-b$—double-check the formula sign! Transferable strategy: write $a$, $b$, $c$ clearly before plugging in. Great precision—you've got this!
10. Solve by taking square roots: $(2x-1)^2=25$.
    Answer: $x=3$ or $x=-2$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving (2x-1)² = 25 by taking square roots: take the square root of both sides to get 2x - 1 = ±5. This gives us two equations: 2x - 1 = 5 (so 2x = 6, thus x = 3) and 2x - 1 = -5 (so 2x = -4, thus x = -2). The ± symbol ensures we capture both solutions! Choice B correctly gives both solutions x = 3 or x = -2 by properly applying the square root method and remembering the ± when taking square roots. Choice A forgets the negative square root and only finds x = 3, missing half the solutions—a common error when students forget that √25 = ±5! Complex solution recognition: before solving, check if you'll get real or complex solutions. For (2x-1)² = 25, since 25 is positive, we get real solutions. Quick method decision tree: IS it already (something) squared = number? Take square roots. For (2x-1)² = 25, apply square roots immediately: 2x - 1 = ±5, giving x = 3 or x = -2. This decision process takes 3 seconds and saves minutes!
11. Solve by completing the square: $x^2+2x+5=0$. (Give solutions in $a\pm bi$ form.)
    Answer: $x=-1\pm 2i$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving x² + 2x + 5 = 0 by completing the square: move the constant to the right side to get x² + 2x = -5. To complete the square, take half of the x-coefficient (which is 1), square it (getting 1), and add to both sides: x² + 2x + 1 = -5 + 1 = -4. This gives (x + 1)² = -4. Taking square roots: x + 1 = ±2i, so x = -1 ± 2i. Choice A correctly completes the square to get (x + 1)² = -4, then takes square roots to find x = -1 ± 2i, properly handling the negative value under the square root. Choice B incorrectly has +1 as the real part instead of -1, perhaps solving x - 1 = ±2i instead of x + 1 = ±2i—remember when (x + 1)² = -4, we get x + 1 = ±2i, so x = -1 ± 2i! Complex solution recognition: When completing the square yields (expression)² = negative number, you'll get complex solutions. For x² + 2x + 5 = 0: complete to (x + 1)² = -4, so x + 1 = ±2i, giving x = -1 ± 2i. The negative on the right side of (x + 1)² = -4 signals complex solutions!
12. Use the quadratic formula to solve: $2x^2 - 3x - 2 = 0$.​
    Answer: $x=\dfrac{3\pm\sqrt{25}}{4}$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For 2x squared - 3x - 2 = 0, a=2, b=-3, c=-2; discriminant=9+16=25; x=[3 ± 5]/4, so x=2 or x=-0.5. The formula handles the leading coefficient easily. Choice A correctly applies the formula with -b as positive 3 and proper denominator 4. Choice B forgets the sign of -b, flipping it to negative—double-check signs! Formula tip: write it out fully before plugging in values to avoid errors. Practice on quadratics with a≠1 to get comfortable.
13. Solve by factoring: $x^2-7x+10=0.$
    Answer: $x=2,\;5$ This question assesses solving quadratics by factoring, where you find two numbers that multiply to c and add to b, then use the zero product property. For $x^2 -7x +10=0$, the numbers are -2 and -5 (multiply to 10, add to -7), so $(x-2)(x-5)=0$, giving $x=2$ or $5$—nice work! Factoring is efficient when the quadratic factors neatly over integers, as it does here. Choice B correctly lists the positive roots from factoring. Choice C might tempt if you mistakenly use positive factors but assign negative signs incorrectly—always check by expanding back to the original. For transferable skills, try factoring mentally first; if it doesn't work in seconds, switch to the quadratic formula. You're building a strong toolkit—keep going!
14. Solve by taking square roots: $(2x+1)^2 = 36$
    Answer: $x=\dfrac{5}{2}$ or $x=-\dfrac{7}{2}$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's $x^2 = \text{number}$, inspect or take square roots ($x = \pm \sqrt{\text{number}}$), (2) if it's $(\text{expression})^2 = \text{number}$, take square roots, (3) if it factors easily (like $x^2 + 5x + 6$), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For $(2x+1)^2 = 36$ by taking square roots: $2x+1 = \pm 6$, so $2x=5$ or $2x=-7$, $x=\frac{5}{2}$ or $x=-\frac{7}{2}$. Choice B correctly solves both branches. Choice A ignores the negative root—always consider both plus and minus! Strategy: isolate the squared term, then apply square roots with $\pm$. You're nailing efficiency—keep it up!
15. Solve by completing the square: $x^2+8x+1=0$
    Answer: $x=-4\pm\sqrt{15}$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For x squared + 8x + 1 = 0 by completing square: move 1, add $(4)^2$=16 to both, (x+4) squared = 15, x = -4 plus or minus sqrt(15). Choice A correctly shifts and takes square root with the right adjustment. Choice B flips the sign—remember, for +bx, you add $(b/2)^2$ and subtract from x! Strategy: halve b, square it, add to both sides. Super effort—completing the square is versatile!
16. Which method is most efficient for solving the quadratic equation $x^2 = 81$?
    Answer: Inspection This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving x squared = 81, this is already in the perfect form x squared = number, so inspection immediately gives x = plus or minus 9. No need for factoring, completing the square, or the quadratic formula—you can see the answer! Choice C (Inspection) correctly identifies the most efficient method because you can instantly recognize that x = plus or minus 9 without any calculations. Choice A (Quadratic formula) would work but requires identifying a = 1, b = 0, c = -81, then calculating x = (0 plus or minus square root of (0 - 4(1)(-81))) divided by 2 = plus or minus square root of 324 divided by 2 = plus or minus 18 divided by 2 = plus or minus 9—way more work for the same answer! Quick method decision tree: IS it already (something) squared = number? Take square roots. DOES it factor in 5 seconds mentally? Factor. NEITHER of above? Quadratic formula (reliable, always works). For x squared = 81, don't use the formula—just see x = plus or minus 9!
17. The solutions to the equation $x^2 + bx + c = 0$ are $x = 3 \pm 2i$. What are the values of $b$ and $c$?
    Answer: $b = -6$ and $c = 13$ If the solutions are x = 3 ± 2i, then x = 3 + 2i and x = 3 - 2i. Using Vieta's formulas: sum of roots = (3 + 2i) + (3 - 2i) = 6 = -b, so b = -6. Product of roots = (3 + 2i)(3 - 2i) = 9 - (2i)² = 9 - 4(-1) = 9 + 4 = 13 = c. Choice B has wrong sign for b. Choice C uses c = 3² - (2i)² = 9 - 4 = 5, forgetting that (2i)² = -4. Choice D confuses the real part with the sum of roots.
18. Solve the quadratic equation and express any complex solutions in $a \pm bi$ form: $x^2 + 6x + 13 = 0$
    Answer: $x = -3 \pm 2i$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's $x$ squared = number, inspect or take square roots ($x = \pm \sqrt{\text{number}}$), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like $x$ squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For $x$ squared + 6x + 13 = 0: discriminant $36 - 52 = -16$ (negative, so complex), quadratic formula gives $x = \frac{-6 \pm \sqrt{-16}}{2} = \frac{-6 \pm 4i}{2} = -3 \pm 2i$. Choice A correctly applies the formula and simplifies to the a plus or minus bi form. Choice B flips the sign of the real part—remember, it's -b over 2a, so negative for positive b! Complex solution recognition: check discriminant first; negative means i appears. You're mastering complex numbers—keep going!
19. Solve and express solutions in $a \pm bi$ form: $x^2 + 4x + 8 = 0$.
    Answer: $x = -2 \pm 2i$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. For solving x squared + 4x + 8 = 0: identify a = 1, b = 4, c = 8. Calculate discriminant: b squared - 4ac = 16 - 32 = -16 (negative, so complex solutions!). Apply formula: x = (-4 plus or minus square root of -16) divided by 2 = (-4 plus or minus 4i) divided by 2 = -2 plus or minus 2i. The negative discriminant signals that i will appear in the answer. Choice A correctly identifies x = -2 plus or minus 2i as the complex solutions in a plus or minus bi form. Choice B incorrectly has x = 2 plus or minus 2i, forgetting that -b/2a = -4/2 = -2, not 2. When b = 4, we get -b = -4 in the numerator, and dividing by 2a = 2 gives -2 for the real part. Complex solution recognition: for x squared + 4x + 8 = 0, discriminant = 16 - 32 = -16 (negative!), so complex solutions: x = (-4 plus or minus square root of -16) divided by 2 = (-4 plus or minus 4i) divided by 2 = -2 plus or minus 2i. The pattern is clear: real part = -b/(2a) = -4/2 = -2, imaginary part = plus or minus square root of |discriminant|/(2a) = plus or minus 4/2 = plus or minus 2.
20. Solve the quadratic equation by factoring: $x^2 - 9x + 20 = 0$.​
    Answer: $x=5$ or $x=4$ This question tests your understanding that quadratic equations can be solved by multiple methods—inspection, taking square roots, factoring, completing the square, and the quadratic formula—and that choosing the most efficient method depends on the equation's form. Method selection guide: (1) if it's x squared = number, inspect or take square roots (x = plus or minus square root of number), (2) if it's (expression) squared = number, take square roots, (3) if it factors easily (like x squared + 5x + 6), factor and use zero product property, (4) if it doesn't factor nicely or you're unsure, use the quadratic formula x = (-b plus or minus square root of (b squared - 4ac)) divided by (2a)—it always works!, (5) if asked to derive the formula, complete the square on general form. Matching method to form saves time and reduces errors! For solving x squared - 9x + 20 = 0 by factoring: look for two numbers that multiply to 20 and add to -9, which are -4 and -5, so (x - 4)(x - 5) = 0, then x = 4 or x = 5. Factoring works here because the quadratic factors nicely over integers. Choice B correctly identifies the solutions as x=5 or x=4 by properly factoring and applying the zero product property. Choice C is tempting but fails because it uses positive factors instead of negative, leading to negative solutions which don't satisfy the equation—always check by plugging back in! To master factoring, practice finding factor pairs: for ac=20, pairs are 1 and 20, 2 and 10, 4 and 5—then pick the pair that adds to b with correct signs. Quick method decision: if the constant and coefficient suggest easy factors (like here, 20 and -9), factor first; otherwise, go to quadratic formula for reliability.

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