Question

im2 midterm study guide #3
test overview 9
question 7
solve the equation \\(\sqrt{6y + 15} - 4 = y\\)
answer
additional solution no solution
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Explanation:

Step1: Isolate the square root

Add 4 to both sides of the equation $\sqrt{6y + 15}-4 = y$ to get $\sqrt{6y + 15}=y + 4$.

Step2: Square both sides

Square both sides: $(\sqrt{6y + 15})^2=(y + 4)^2$, which simplifies to $6y+15=y^{2}+8y + 16$.

Step3: Rearrange into quadratic form

Rearrange the equation to $y^{2}+8y + 16-6y - 15 = 0$, so $y^{2}+2y + 1 = 0$.

Step4: Solve the quadratic equation

Factor the quadratic: $(y + 1)^2=0$, so $y=-1$.

Step5: Check the solution

Substitute $y = - 1$ into the original equation: Left side $\sqrt{6(-1)+15}-4=\sqrt{9}-4 = 3 - 4=-1$, right side is $-1$. So $y=-1$ is a valid solution.

Answer:

$y = - 1$