An equation is shown below. \sqrt{2x + 7} = x - 4 If x satisfies the equation above, what … — SAT

An equation is shown below.

\sqrt{2x + 7} = x - 4

x

satisfies the equation above, what is the value of

x + 3

To solve the equation, square both sides to obtain

2x + 7 = x^2 - 8x + 16

. Rearranging terms gives the quadratic equation

x^2 - 10x + 9 = 0

, which factors as

(x - 9)(x - 1) = 0

. This yields potential solutions of

x = 9

and

x = 1

. Testing these in the original equation shows that

x = 9

is a valid solution because

\sqrt{25} = 5

, but

x = 1

is extraneous because

\sqrt{9} \neq -3

. Thus, the only real solution is

x = 9

. Evaluating

x + 3

for this solution gives

9 + 3 = 12

Square both sides of the equation to eliminate the radical.

2x + 7 = (x - 4)^2

, which expands to

2x + 7 = x^2 - 8x + 16

Squaring both sides is the standard method to solve radical equations.

Rearrange the terms to set the quadratic equation to zero.

x^2 - 10x + 9 = 0

This allows us to solve the quadratic equation by factoring.

Factor the quadratic equation.

(x - 9)(x - 1) = 0

, giving potential solutions of

x = 9

and

x = 1

Factoring helps find the roots of the quadratic equation.

Verify each potential solution in the original equation to check for extraneous roots.

x = 9

is valid because

\sqrt{2(9)+7} = 9-4

is true (

5=5

x = 1

is extraneous because

\sqrt{2(1)+7} = 1-4

is false (

3 \neq -3

Squaring both sides can introduce extraneous solutions that do not satisfy the original equation.

Substitute the valid solution

x = 9

into the expression

x + 3

9 + 3 = 12

The question asks for the value of the expression

x + 3

rather than just

x

Solving radical equations and checking for extraneous solutions

Question