The quadratic equation x^2 - 8x - 9 = 0 can be written in the equivalent form (x - a)^2 - … — SAT

The quadratic equation $x^2 - 8x - 9 = 0$ can be written in the equivalent form $(x - a)^2 - b = 0$ , where $a$ and $b$ are positive constants. What is the value of $a + b$ ?

A
11
B
13
29Answer
D
21

Answer

The value of

a + b

is 29.

Completing the square on the quadratic equation

x^2 - 8x - 9 = 0

yields

(x - 4)^2 - 25 = 0

. Comparing this to the form

(x - a)^2 - b = 0

shows that

a = 4

and

b = 25

. The sum of these values is

4 + 25 = 29

Step-by-Step Solution

Identify the coefficient of the linear term and find half of its value.

The coefficient of the linear term

-8x

-8

. Half of

-8

-4

This is the first step in completing the square.

Square the value obtained in the first step and add/subtract it in the equation to form a perfect square trinomial.

(-4)^2 = 16

. The equation becomes

(x^2 - 8x + 16) - 16 - 9 = 0

Adding and subtracting

16

maintains the equality while allowing us to group the first three terms as a perfect square.

Rewrite the perfect square trinomial and combine the remaining constant terms.

(x - 4)^2 - 25 = 0

This simplifies the equation into the desired equivalent form

(x - a)^2 - b = 0

Compare the equation to the target form

(x - a)^2 - b = 0

to identify the constants

a

and

b

, and calculate

a + b

a = 4

and

b = 25

. Therefore,

a + b = 4 + 25 = 29

This answers the question by finding the sum of the positive constants

a

and

b

Key Concept

Completing the square to rewrite a quadratic equation

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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