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Chapter 0: Problem 51

Factor each perfect square trinomial. $$x^{2}-14 x+49$$

Short Answer

Expert verified
The factored form of the trinomial is \((x - 7)^2\).

Step by step solution

01

Identify the form of the trinomial

The given trinomial is \(x^{2}-14x+49\). It can be noticed that this trinomial is in the form of \(a^2 - 2ab + b^2\).
02

Identify the values of a and b

Compare the trinomial with the form \(a^2 - 2ab + b^2\) to identify the values of \(a\) and \(b\). Here, \(a = x\) (since \(a^2 = x^2\)), and \(b = 7\) (since \(b^2 = 49\)). Our task is to check whether \(2ab = -14x\) or not. With \(a = x\) and \(b = 7\), we have \(2ab = 2*x*7 = 14x\), so indeed our original trinomial is in the form of a perfect square trinomial.
03

Apply the factoring formula

When the trinomial is in the form \(a^2 - 2ab + b^2\), it can be factored into \((a - b)^2\). Substituting the values of \(a\) and \(b\) in this formula, we get \((x - 7)^2\).

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Most popular questions from this chapter

Why must \(a\) and \(b\) represent nonnegative numbers when we write \(\sqrt{a} \cdot \sqrt{b}=\sqrt{a b} ?\) Is it necessary to use this restriction in the case of \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{a b} ?\) Explain.

How do the whole numbers differ from the natural numbers?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using my calculator, I determined that \(6^{7}=279,936,\) so 6 must be a seventh root of \(279,936\).

Determine whether each statement is trueor false. If the statement is false, make the necessary change(s) toproduce a true statement. \(x^{4}-16\) is factored completely as \(\left(x^{2}+4\right)\left(x^{2}-4\right)\)

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(8 x^{-6} y^{3}\right)^{\frac{1}{3}}\left(x^{\frac{5}{6}} y^{-\frac{1}{3}}\right)^{6} $$
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