Question

selena completes the following steps to solve the equation 6 - x = √x. equation: 6 - x = √x. step 1: (6 - x)² = x. step 2: 36 - 12x + x² = x. step 3: x² - 13x + 36 = 0. step 4: (x - 4)(x - 9) = 0. step 5: x - 4 = 0 so x = 4, or x - 9 = 0 so x = 9. step 6: the solutions are 4 or 9. if selena made a mistake in her work, in which step did it occur? a. selena made no mistakes. her solutions are correct. b. selenas mistake was in step 1. when she squared both sides of the equation to eliminate the square root, she should have changed the value on the left side to +. c. selenas mistake was in step 4. the quadratic equation from step 3 cant be factored into the two binomials shown in step 4. d. selenas mistake was in step 6. she should have substituted the value 9 back into the original equation to determine that it is an extraneous solution.

Explanation:

Step1: Analyze step - 1

In step 1, when squaring both sides of the equation \(6 - x=\sqrt{x}\), we get \((6 - x)^2=x\). This is a correct operation based on the property \(a = b\Rightarrow a^{2}=b^{2}\) for real - valued \(a\) and \(b\).

Step2: Analyze step - 2

Expanding \((6 - x)^2\) using the formula \((a - b)^2=a^{2}-2ab + b^{2}\) where \(a = 6\) and \(b=x\), we have \(36-12x+x^{2}=x\), which is correct.

Step3: Analyze step - 3

Rearranging the equation \(36-12x+x^{2}=x\) to the standard quadratic form \(x^{2}-13x + 36=0\) is correct.

Step4: Analyze step - 4

Factoring the quadratic equation \(x^{2}-13x + 36=0\) into \((x - 4)(x - 9)=0\) is correct since \((x - 4)(x - 9)=x^{2}-9x-4x + 36=x^{2}-13x + 36\).

Step5: Analyze step - 5

Setting each factor equal to zero, \(x-4 = 0\) gives \(x = 4\) and \(x - 9=0\) gives \(x = 9\), which is the correct way to solve a factored quadratic equation.

Step6: Analyze step - 6

We need to check for extraneous solutions by substituting the values back into the original equation \(6 - x=\sqrt{x}\). When \(x = 4\), \(6-4=\sqrt{4}\), \(2 = 2\) (a valid solution). When \(x = 9\), \(6-9=\sqrt{9}\), \(- 3=3\) (not valid). The error is that in step 6, Selena did not check for extraneous solutions properly. She should have substituted the values back into the original equation to determine if they are valid.

Answer:

D. Selena's mistake was in step 6. She should have substituted the value 9 back into the original equation to determine that it is an extraneous solution.