Problem 53 - College Algebra

Question

Solve each equation in by making an appropriate substitution. $$ (x-5)^{2}-4(x-5)-21=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x=12 \) and \( x=2 \)

Step 1: Substitute

Substitute \( u \) for \( (x-5) \), the quadratic equation then becomes: \( u^{2}-4u-21=0 \)

Step 2: Solve the resulting quadratic equation

Perform factoring to solve the quadratic equation. So, factoring \( u^{2}-4u-21=0 \) the factors out to be \( (u-7)(u+3) = 0 \)

Step 3: Solve for the variable\( u \)

Set each factor equal to zero and solve for \( u \) so you have \( u-7=0 \) and \( u+3=0 \). Solving these equations we get values of \( u \) such as \( u=7 \) and \( u=-3 \)

Step 4: Return to the original variable \( x \)

Replace the \( u \) in both solutions with the original expression it replaced, which is \( (x-5) \). Then, solve for \( x \) by solving the equations \( (x-5)=7 \) and \( (x-5)=-3 \). This leads us to \( x=7+5=12 \) and \( x=-3+5=2 \)