Algebra 2 Quiz: Solve Quadratics By Multiple Methods

20 questions · exam conditions

0:00

Solve Quadratics By Multiple MethodsQuestion 1 of 20

Solve using the quadratic formula: $2x^2-3x-7=0$ .

x=\dfrac{3\pm\sqrt{65}}{4}

x=\dfrac{-3\pm\sqrt{65}}{4}

x=\dfrac{3\pm\sqrt{17}}{4}

x=\dfrac{3\pm\sqrt{65}}{2}

← Back to quizzes

Algebra 2 Quiz

Algebra 2 Quiz: Solve Quadratics By Multiple Methods

Practice Solve Quadratics By Multiple Methods in Algebra 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solve Quadratics By Multiple Methods, giving you a quick way to practice the rules, question types, and explanations that matter most for Algebra 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Solve using the quadratic formula: $2x^2-3x-7=0$ .

$x=\dfrac{3\pm\sqrt{65}}{4}$ (correct answer)
$x=\dfrac{-3\pm\sqrt{65}}{4}$
$x=\dfrac{3\pm\sqrt{17}}{4}$
$x=\dfrac{3\pm\sqrt{65}}{2}$

Explanation: 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 2x² - 3x - 7 = 0 by quadratic formula: identify a = 2, b = -3, c = -7. Calculate discriminant: b² - 4ac = (-3)² - 4(2)(-7) = 9 + 56 = 65 (positive, so two real solutions!). Apply formula: x = (3 ± √65)/(2·2) = (3 ± √65)/4. Note that -b = -(-3) = 3, which is positive! Choice A correctly applies the quadratic formula with proper sign handling: -b = -(-3) = +3, giving x = (3 ± √65)/4. Choice B incorrectly uses -3 instead of 3 in the numerator, forgetting that when b = -3, then -b = +3—a common sign error in the quadratic formula! Quick method decision tree: For 2x² - 3x - 7 = 0, factoring would be messy (need to factor 2·(-7) = -14), so jump straight to the quadratic formula. With a = 2, b = -3, c = -7, we get x = (3 ± √65)/4. The formula always works and gives exact answers!

Question 2

Which equation is factoring most efficient for solving?

A. $x^2+5x+6=0$ B. $x^2+5x+1=0$ C. $2x^2+3x-7=0$ D. $x^2+4x+8=0$

$x^2+5x+6=0$ (correct answer)
$x^2+5x+1=0$
$2x^2+3x-7=0$
$x^2+4x+8=0$

Explanation: This question examines which quadratic is most efficient to solve by factoring, based on whether it factors easily over integers without fractions or complex numbers. For x^2 +5x +6=0, it factors to (x+2)(x+3)=0 quickly, as 2 and 3 multiply to 6 and add to 5—ideal! Others like B don't factor nicely (discriminant 21, irrational), C has a leading coefficient of 2 complicating factors, and D has complex roots. Choice A is correctly identified as easiest for factoring. Choice B might tempt if you overlook that 5x doesn't split into integers multiplying to 1—check factors of c that add to b. Scan for integer factors first; if not, pivot to formula. You're getting better at this—keep practicing for speed!

Question 3

The quadratic $3x^2 + 12x + 15 = 0$ has solutions of the form $a \pm bi$ . What are the values of $a$ and $b$ ?

$a = -2$ and $b = 1$ (correct answer)
$a = 2$ and $b = 1$
$a = -2$ and $b = \sqrt{3}$
$a = 4$ and $b = \sqrt{3}$

Explanation: Using the quadratic formula: x = (-12 ± √(144 - 180))/6 = (-12 ± √(-36))/6 = (-12 ± 6i)/6 = -2 ± i. Therefore a = -2 and b = 1. Choice B has the wrong sign for a. Choice C incorrectly calculates b as √3 instead of 1. Choice D has both wrong values, possibly from calculation errors in the quadratic formula.

Question 4

The solutions to the equation $x^2 + bx + c = 0$ are $x = 3 \pm 2i$ . What are the values of $b$ and $c$ ?

$b = -6$ and $c = 13$ (correct answer)
$b = 6$ and $c = 13$
$b = -6$ and $c = 5$
$b = -3$ and $c = 4$

Explanation: 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.

Question 5

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?

$k = 7$ ; quadratic formula is most efficient due to the discriminant being perfect square
$k = 7$ ; factoring is most efficient since $x^2 - 8x + 7 = (x - 1)(x - 7)$ (correct answer)
$k = 16$ ; taking square roots is most efficient since the original is already perfect square
$k = -7$ ; completing the square is most efficient since the coefficient of $x$ is even

Explanation: 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.

Question 6

A quadratic equation $ax^2 + bx + c = 0$ has discriminant $\Delta = 36$ . If the equation is solved using the quadratic formula, and the solutions are $x = \frac{-4 \pm 6}{10}$ , what is the original equation?

$5x^2 - 4x - 1 = 0$
$10x^2 + 8x - 2 = 0$
$5x^2 + 4x - 1 = 0$ (correct answer)
$10x^2 - 8x + 2 = 0$

Explanation: When you're given solutions from the quadratic formula and need to work backwards to find the original equation, you're essentially reverse-engineering the quadratic formula process. The quadratic formula is

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

. Comparing this to the given solutions

x = \frac{-4 \pm 6}{10}

, you can identify the components: the numerator tells us that

-b = -4

(so

b = 4

) and

\sqrt{\Delta} = \sqrt{36} = 6

. The denominator shows us that

2a = 10

, so

a = 5

. Now you need to find

c

. Since the discriminant

\Delta = b^2 - 4ac = 36

, substitute the known values:

4^2 - 4(5)c = 36

. This gives us

16 - 20c = 36

, so

-20c = 20

, and

c = -1

. Therefore, the original equation is

5x^2 + 4x - 1 = 0

, which is choice C. Let's check why the other options are wrong: Choice A has

b = -4

instead of

b = 4

, which would give solutions with

+4

in the numerator, not

-4

. Choice B has

a = 10

, making the denominator

2a = 20

, not 10. Choice D has both

b = -8

(wrong sign and magnitude) and

c = 2

(wrong sign), leading to completely different solutions. Strategy tip: When working backwards from quadratic formula solutions, always match the pattern

\frac{-b \pm \sqrt{\Delta}}{2a}

term by term. The denominator gives you

a

, the first part of the numerator gives you

b

, and use the discriminant formula to find

c

Question 7

A quadratic equation has the form $ax^2 + bx + c = 0$ where $a = 3$ , $b = -12$ , and $c = k$ . If the equation can be solved by factoring into the form $3(x - p)(x - q) = 0$ where $p$ and $q$ are integers, and one solution is $x = 5$ , what is the value of $k$ ?

$k = -15$ (correct answer)
$k = 15$
$k = -45$
$k = 45$

Explanation: If one solution is x = 5, then (x - 5) is a factor. Since the equation factors as 3(x - p)(x - q) = 0, we have 3x² - 12x + k = 3(x - 5)(x - q) for some integer q. From the coefficient of x, we get -12 = 3(-5 - q), so -4 = -5 - q, giving q = -1. Therefore, 3x² - 12x + k = 3(x - 5)(x + 1) = 3(x² - 4x - 5) = 3x² - 12x - 15. Thus k = -15. Choice B gives the opposite sign. Choice C results from incorrectly multiplying 3 × 5 × (-3) = -45. Choice D results from incorrectly multiplying 3 × 5 × 3 = 45.

Question 8

A student attempts to solve $2x^2 - 12x + 10 = 0$ by factoring. After factoring out the GCF, what quadratic expression remains, and what is the most appropriate next method to find the solutions?

$x^2 - 6x + 5 = 0$ ; factor as $(x - 1)(x - 5) = 0$ (correct answer)
$x^2 - 6x + 5 = 0$ ; use quadratic formula since it doesn't factor nicely
$2x^2 - 12x + 10 = 0$ ; complete the square on the original equation
$x^2 - 3x + 2.5 = 0$ ; factor as $(x - 1.5)(x - 2) = 0$

Explanation: First factor out GCF of 2: 2(x² - 6x + 5) = 0, so x² - 6x + 5 = 0. This factors nicely as (x - 1)(x - 5) = 0, giving x = 1 or x = 5. Choice B correctly identifies the remaining quadratic but incorrectly suggests it doesn't factor. Choice C doesn't factor out the GCF. Choice D incorrectly divides the middle term and constant by 2.

Question 9

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 $m$ ?

$m = 10$
$m = 14$
$m = 9.5$
$m = 11$ (correct answer)

Explanation: 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.

Question 10

Which method is most efficient for solving the equation

$(x-3)^2 = 11$

and then finding the solutions?

Factoring
Quadratic formula
Taking square roots (correct answer)
Completing the square

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!

Question 11

Solve using the quadratic formula and simplify: $2x^2-3x-7=0.$

$x=\dfrac{3\pm\sqrt{17}}{2}$
$x=\dfrac{-3\pm\sqrt{65}}{4}$
$x=\dfrac{3\pm\sqrt{65}}{4}$ (correct answer)
$x=\dfrac{3\pm\sqrt{121}}{4}$

Explanation: This question focuses on using the quadratic formula for equations that don't factor easily, reminding you to identify a, b, c correctly and simplify the result. The formula is

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

; for

2x^2 - 3x - 7 = 0

a=2

b=-3

c=-7

, discriminant=

9+56=65

, so

x=\frac{3 \pm \sqrt{65}}{4}

—excellent! This method always works, especially for non-integer coefficients or tricky factoring. Choice A accurately applies the formula, noting -b becomes +3. Choice C forgets to flip the sign of b, leading to -3 instead—double-check that -b step every time! Always calculate the discriminant first to predict real or complex roots. With practice, you'll choose and apply methods confidently— you've got this!

Question 12

Solve the quadratic equation. (Choose any method.) $x^2+3x-1=0.$

$x=\dfrac{-3\pm\sqrt{13}}{2}$ (correct answer)
$x=\dfrac{3\pm\sqrt{13}}{2}$
$x=\dfrac{-3\pm\sqrt{5}}{2}$
$x=-3\pm\sqrt{13}$

Explanation: This question allows any method to solve a quadratic, testing flexibility and accuracy in finding roots for non-factorable equations. For

x^2 + 3x - 1 = 0

, quadratic formula works well:

discriminant=9+4=13

x=\frac{-3 \pm \sqrt{13}}{2}

—nice and precise! Factoring isn't easy here, as it doesn't split neatly. Choice A correctly gives the solutions with the negative sign from

-b

. Choice B flips the sign incorrectly—recall

-b

for positive 3 becomes -3 in numerator. Try different methods on the same equation to verify consistency. This versatility will serve you well—excellent effort!

Question 13

Solve the quadratic equation by factoring: $x^2 - 9x + 20 = 0$ .

$x=-5$ or $x=-4$
$x=5$ or $x=4$ (correct answer)
$x=4$ only
$x=9$ or $x=20$

Explanation: 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.

Question 14

Solve by taking square roots: $(x - 4)^2 = 20$ .

$x = 4 \pm \sqrt{20}$ (correct answer)
$x = 4 + \sqrt{20}$
$x = \pm \sqrt{20}$
$x = -4 \pm \sqrt{20}$

Explanation: 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 - 4) squared = 20 by taking square roots: take the square root of both sides to get x - 4 = plus or minus square root of 20, then add 4 to both sides to isolate x, giving x = 4 plus or minus square root of 20. The key is remembering the plus or minus when taking square roots! Choice A correctly solves by taking square roots and includes both solutions: x = 4 plus square root of 20 and x = 4 minus square root of 20. Choice B incorrectly has x = -4 plus or minus square root of 20, which would come from (x + 4) squared = 20, not (x - 4) squared = 20. When you have (x - a) squared = b, the solution is x = a plus or minus square root of b. The 'a' value comes from what's being subtracted inside the parentheses, and you ADD it back when isolating x. This pattern works every time: (x - 4) squared = 20 gives x = 4 plus or minus square root of 20, while (x + 4) squared = 20 would give x = -4 plus or minus square root of 20.

Question 15

Which method is most efficient for solving $(2x-1)^2=25$ ?

A. factoring B. taking square roots C. completing the square D. quadratic formula

completing the square
taking square roots (correct answer)
quadratic formula
factoring

Explanation: This question asks for the most efficient method to solve a quadratic in (expression)^2 = number form, highlighting method selection based on equation structure. Taking square roots is quickest: for (2x-1)^2=25, 2x-1=±5, then 2x=6 or -4, x=3 or -2—efficient and direct! Expanding to standard form would work but adds unnecessary steps. Choice B correctly picks taking square roots for this form. Choice A, factoring, tempts if you expand first, but it's less efficient—stick to the given form. Use the decision tree: if squared equals constant, square root first. You're sharpening your skills—great job choosing wisely!

Question 16

A quadratic equation can be written as $(x - 4)^2 = 25$ . Which of the following represents all solutions when this equation is solved using a different method?

$x = \frac{8 \pm \sqrt{164}}{2}$ using the quadratic formula on $x^2 - 8x - 9 = 0$
$x = \frac{-8 \pm \sqrt{100}}{2}$ using the quadratic formula on $x^2 + 8x + 9 = 0$
$(x - 9)(x + 1) = 0$ when factoring $x^2 - 8x - 9 = 0$ (correct answer)
$(x + 9)(x - 1) = 0$ when factoring $x^2 + 8x - 9 = 0$

Explanation: From (x - 4)² = 25, we get x - 4 = ±5, so x = 4 ± 5, giving x = 9 or x = -1. Expanding (x - 4)² = 25 gives x² - 8x + 16 = 25, so x² - 8x - 9 = 0. This factors as (x - 9)(x + 1) = 0, confirming x = 9 or x = -1. Choice A has the wrong discriminant (should be 64 + 36 = 100). Choice B has wrong signs in the quadratic. Choice D factors a different quadratic with wrong solutions.

Question 17

Solve by taking square roots: $(x-3)^2 = 5$ .

$x=3\pm\sqrt{5}$ (correct answer)
$x=-3\pm\sqrt{5}$
$x=3\pm 5$
$x=\sqrt{5}-3$ only

Explanation: 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.

Question 18

Solve and express solutions in $a \pm bi$ form: $x^2 + 4x + 8 = 0$ .

$x = -4 \pm 2i$
$x = -2 \pm i$
$x = 2 \pm 2i$
$x = -2 \pm 2i$ (correct answer)

Explanation: 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.

Question 19

For the equation $x^2 - 6x + 10 = 0$ , the discriminant is negative. Solve and write the solutions in $a \pm bi$ form.

$x = 3 \pm 2i$
$x = -3 \pm i$
$x = 6 \pm i$
$x = 3 \pm i$ (correct answer)

Explanation: 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 - 6x + 10 = 0 with negative discriminant: identify a = 1, b = -6, c = 10. Calculate discriminant: b squared - 4ac = 36 - 40 = -4 (negative, so complex solutions!). Apply formula: x = (6 plus or minus square root of -4) divided by 2 = (6 plus or minus 2i) divided by 2 = 3 plus or minus i. The negative discriminant signals that i will appear in the answer. Choice A correctly identifies x = 3 plus or minus i as the complex solutions in a plus or minus bi form. Choice B incorrectly has x = -3 plus or minus i, which would come from using b = 6 instead of b = -6 in the formula. When b = -6, we get -b = -(-6) = 6 in the numerator, giving x = (6 plus or minus 2i) divided by 2 = 3 plus or minus i. Complex solution recognition: before solving, check the discriminant b squared - 4ac. Negative means two complex solutions (use formula or completing square, write as a plus or minus bi). For x squared - 6x + 10 = 0: discriminant = 36 - 40 = -4 (negative!), so complex solutions: x = (6 plus or minus square root of -4) divided by 2 = (6 plus or minus 2i) divided by 2 = 3 plus or minus i.

Question 20

Solve the equation by any method. If the solutions are complex, write them in $a\pm bi$ form:

$3x^2+6x+5=0$

$x=-1\pm\dfrac{\sqrt{6}}{6}i$
$x=1\pm\dfrac{\sqrt{6}}{3}i$
$x=-1\pm\dfrac{\sqrt{6}}{3}$
$x=-1\pm\dfrac{\sqrt{6}}{3}i$ (correct answer)

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!