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Chapter 1: Problem 46

Find the value(s) of \(x\) for which \(f(x)=g(x)\). $$f(x)=x^{2}+2 x+1, \quad g(x)=7 x-5$$

Short Answer

Expert verified
The values of \(x\) for which \(f(x)\) equals \(g(x)\) are \(x=3\) and \(x=2\).

Step by step solution

01

Set the two functions equal to each other

To find the values of \(x\) that make \(f(x)\) equal to \(g(x)\), the equation is set as \(f(x)=g(x)\). This translates to \(x^{2}+2 x+1=7 x-5\).
02

Rearrange the equation

Next, we rearrange the equation to set it to zero. This helps in finding the solutions by factoring. We subtract \(7x\) from both sides and add 5 to both sides to get \(x^{2}-5x+6=0\).
03

Factor the quadratic expression

Factoring the quadratic expression \(x^{2}-5x+6=0\) gives us \((x-3)(x-2)=0\).
04

Find the roots

Setting each factor equal to zero and solving for \(x\), we get \(x=3\) and \(x=2\).

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