Question

Solve the square root equation 7x−12−−−−−−√=x
7
𝑥
−
12
=
𝑥
.(1 point)
Responses

x=−4
𝑥
=
−
4
and x=−3
𝑥
=
−
3
x equals negative 4 and x equals negative 3

x=32
𝑥
=
3
2
x equals Start Fraction 3 over 2 End Fraction

x=2
𝑥
=
2
x equals 2

x=3
𝑥
=
3
and x=4

Answer 1

To solve the equation \( \sqrt{7x - 12} = x \), we will start by squaring both sides of the equation to eliminate the square root.

1. Square both sides:
\[
(\sqrt{7x - 12})^2 = x^2
\]
This simplifies to:
\[
7x - 12 = x^2
\]

2. Rearrange the equation to form a standard quadratic equation:
\[
x^2 - 7x + 12 = 0
\]

3. Factor the quadratic:
\[
(x - 3)(x - 4) = 0
\]

4. Set each factor equal to zero:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]

5. Verify both solutions in the original equation:
For \( x = 3 \):
\[
\sqrt{7(3) - 12} = \sqrt{21 - 12} = \sqrt{9} = 3
\]
(Valid solution)

For \( x = 4 \):
\[
\sqrt{7(4) - 12} = \sqrt{28 - 12} = \sqrt{16} = 4
\]
(Valid solution)

Thus, the solutions to the equation \( \sqrt{7x - 12} = x \) are \( x = 3 \) and \( x = 4 \).

The correct response is:
- \( x = 3 \) and \( x = 4 \).