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Chapter 1: Problem 53

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

Short Answer

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

Step by step solution

01

Substitute

Substitute \( u \) for \( (x-5) \), the quadratic equation then becomes: \( u^{2}-4u-21=0 \)
02

Solve the resulting quadratic equation

Perform factoring to solve the quadratic equation. So, factoring \( u^{2}-4u-21=0 \) the factors out to be \( (u-7)(u+3) = 0 \)
03

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 \( u=-3 \)
04

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 \( x=-3+5=2 \)

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Most popular questions from this chapter

What is a rational inequality?

In Exercises \(54-56,\) perform the indicated operations and write the result in standard form. $$\frac{1+i}{1+2 i}+\frac{1-i}{1-2 i}$$

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{5 i}{2-i}$$

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{2 i}{1+i}$$

Describe similarities and differences between the solutions of $$ (x-2)(x+5) \geq 0 \text { and } \frac{x-2}{x+5} \geq 0 $$
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