Question

what is the solution to the equation?
\\(\sqrt{x} + 6 = x\\)
a. 4
b. 9
c. 4 and 9
d. no solution

Explanation:

Step1: Isolate the square root

Subtract 6 from both sides of the equation \(\sqrt{x}+6 = x\) to get \(\sqrt{x}=x - 6\).

Step2: Square both sides

Square both sides to eliminate the square root: \((\sqrt{x})^2=(x - 6)^2\), which simplifies to \(x=x^{2}-12x + 36\).

Step3: Rearrange into quadratic equation

Rearrange the equation to standard quadratic form: \(x^{2}-13x + 36 = 0\).

Step4: Factor the quadratic

Factor the quadratic: \((x - 4)(x - 9)=0\).

Step5: Solve for x

Set each factor equal to zero: \(x - 4 = 0\) gives \(x = 4\), and \(x - 9 = 0\) gives \(x = 9\).

Step6: Check solutions

• For \(x = 4\): Substitute into the original equation \(\sqrt{4}+6=2 + 6 = 8

eq4\), so \(x = 4\) is extraneous.

• For \(x = 9\): Substitute into the original equation \(\sqrt{9}+6=3 + 6 = 9\), which is true.

Answer:

B. 9