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Chapter 1: Problem 157

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

Short Answer

Expert verified
The solutions to the equation \(x^{2}+6x+8=0\) are \(x=-2\) and \(x=-4\).

Step by step solution

01

Write the equation in standard form

A quadratic equation has the standard form \(ax^{2}+bx+c=0\). Therefore, our equation, \(x^{2}+6x+8=0\) is already in standard form.
02

Write the equation missing its constant term

By writing the quadratic equation without the constant term we get \(x^{2} + 6x = -8\). This is done in order to create a perfect square trinomial.
03

Complete the square

Completing the square involves inserting an additional term \((b/2)^{2}\) that completes the square trinomial. The \(b\) from equation \(ax^{2}+bx+c=0\) is the coefficient of the linear \(x\), hence divide 6 by 2 and square the result to get 9 (i.e., \((3)^2 = 9\)). Therefore, add 9 to both sides to maintain the balance of the equation. This will give: \(x^{2}+6x+9=-8+9\). Simplifying the right-hand side gives \(x^{2}+6x+9=1\). Here, \(x^{2}+6x+9\) is a perfect square trinomial.
04

Solve the resulting equation

The perfect square trinomial can be written as \((x+3)^{2}=1\). Taking square root on both sides gives \(x+3=\pm\sqrt{1}\), hence we have two solutions \(x+3=1\) and \(x+3=-1\). Solving for \(x\) in both equations gives \(x=-2, -4\).

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Most popular questions from this chapter

In Exercises \(103-104,\) use the graph of \(y=|4-x|\) to solve each inequality. $$ |4-x|<5 $$

Will help you prepare for the material covered in the next section. If \(-8\) is substituted for \(x\) in the equation \(5 x^{\frac{2}{3}}+11 x^{\frac{1}{3}}+2=0\) is the resulting statement true or false?

Will help you prepare for the material covered in the first section of the next chapter. Here are two sets of ordered pairs: $$ \begin{aligned} &\text { set } 1:\\{(1,5),(2,5)\\}\\\ &\operatorname{set} 2:\\{(5,1),(5,2)\\} \end{aligned} $$ In which set is each x@coordinate paired with only one y@coordinate?

Find all values of \(x\) satisfying the given conditions. $$ \begin{aligned} &y_{1}=-x^{2}+4 x-2, y_{2}=-3 x^{2}+x-1, \text { and } &y_{1}-y_{2}=0 \end{aligned} $$

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is 3000 dollar and it costs 3.00 dollar to produce each package of stationery. The selling price is $5.50 per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?
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