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!

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!

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 (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 $x^2 + 5x + 6 = 0$, factoring is most efficient: numbers 2 and 3 multiply to 6, add to 5, so $(x+2)(x+3)=0$, $x=-2$ or $-3$. Choice C correctly spots the easy integer factors. Choice D assumes squared form, but it's not—check the equation first! Decision tree: factors quickly? Yes—factor. You're sharpening your judgment—excellent!

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 $(x-3)^2 = 11$, taking square roots is most efficient: $x-3 = \pm \sqrt{11}$, so $x=3 \pm \sqrt{11}$—no need for longer methods. Choice C correctly recognizes the squared binomial form, which directly lends itself to square roots for quick solutions. Choice A might tempt if you expand first, but that's unnecessary work—stay in the given form to spot efficiency! Transferable strategy: scan for $(\text{something})^2 = \text{constant}$; if yes, square roots win. Keep honing this— you're getting faster at method choice!

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!

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!

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!

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 (\text{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 $x^2 = 81$ by inspection: recognize it's a perfect square, $9^2 = 81$, so $x = \pm 9$—simple mental math. Choice C correctly includes both positive and negative roots, as squares erase signs. Choice A forgets the negative root—always remember the plus or minus for square roots! Transferable strategy: for $x^2 = k$, if k is positive, two real solutions: $\pm \sqrt{k}$; if zero, one; if negative, complex. Awesome work—inspection builds speed for basics!

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 (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 $3x^2 + 6x + 5 = 0$: discriminant $36 - 60 = -24$ (negative, complex), formula $x = \frac{-6 \pm \sqrt{-24}}{6} = \frac{-6 \pm 2 \sqrt{6} , i}{6} = -1 \pm \frac{\sqrt{6}}{3} , i$. Choice B correctly computes and simplifies the complex form. Choice A omits i—check discriminant sign! Recognition: negative means $a \pm bi$. Impressive handling of complexes— you're advancing!

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)$^2$ = 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 $$x^2 - 4x + 8 = 0$$: discriminant $$16 - 32 = -16$$ (negative, complex), solutions $$x=\frac{4 \pm 4i}{2} = 2 \pm 2i$$. Choice B correctly identifies complex nature and solves accurately. Choice A assumes real by miscalculating discriminant—always compute $b^2 - 4ac$ first! Complex recognition: negative discriminant means two conjugates with i. You're doing fantastic—embrace the imaginaries!