The solutions to the quadratic equation x^2 - 6x - 11 = 0 can be written in the form x = a… — SAT

The solutions to the quadratic equation $x^2 - 6x - 11 = 0$ can be written in the form $x = a \pm \sqrt{b}$ , where $a$ and $b$ are integers. What is the value of $a + b$ ?

A
8
B
17
23Answer
D
43

Answer

Completing the square on the quadratic equation

x^2 - 6x - 11 = 0

yields the solutions

x = 3 \pm \sqrt{20}

. Comparing this to the given form

x = a \pm \sqrt{b}

identifies

a = 3

and

b = 20

. Summing these values gives

a + b = 23

Step-by-Step Solution

Add 11 to both sides of the equation to isolate the variable terms.

x^2 - 6x = 11

Preparing the quadratic equation to complete the square by separating constant terms.

Complete the square by adding the square of half the coefficient of

x

to both sides.

x^2 - 6x + 9 = 11 + 9

, which simplifies to

(x - 3)^2 = 20

Adding

(\frac{-6}{2})^2 = 9

creates a perfect square trinomial on the left side.

Take the square root of both sides and solve for

x

x - 3 = \pm\sqrt{20}

, which gives

x = 3 \pm \sqrt{20}

Taking the square root cancels the exponent and yields two possible solutions.

Compare the solutions to the given form

x = a \pm \sqrt{b}

to determine

a

and

b

, and calculate their sum.

a = 3

b = 20

, so

a + b = 3 + 20 = 23

Matching the template shows that

a

corresponds to

3

and

b

corresponds to

20

Key Concept

Solving quadratic equations by completing the square and matching solutions to a given form.

Alternative Method

Instead of completing the square, you can use the quadratic formula:

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

with

a = 1

b = -6

, and

c = -11

. This yields

x = \frac{6 \pm \sqrt{36 - 4(1)(-11)}}{2} = \frac{6 \pm \sqrt{80}}{2} = 3 \pm \frac{\sqrt{80}}{2} = 3 \pm \sqrt{20}

. Comparing this to the given form yields

a = 3

and

b = 20

, which sums to

23

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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