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Chapter 0: Problem 52

Factor each perfect square trinomial. $$x^{2}-10 x+25$$

Short Answer

Expert verified
\((x - 5)^{2}\

Step by step solution

01

Identify a and b values

In our trinomial, \(a\) would be the square root of the first term, \(x^{2}\) which is \(x\), and \(b\) would be the square root of the third term, 25, which is 5.
02

Verify the fit to the perfect square trinomial form

Perfect square trinomials fit the form \(a^{2}\)-2ab+\(b^{2}\). The term in the middle, -10x, should equal to -2ab = -2*\(x\)*5 = -10x, and it does. So, this verifies that the original expression we have is indeed a perfect square trinomial.
03

Use the Square of a Binomial rule

Using the square of a binomial rule, which states that \((a-b)^{2} = a^{2} - 2ab + b^{2}\), we can rewrite our trinomial. Considering our identified \(a\) and \(b\) values, our original equation \(x^{2} - 10x + 25\) is equal to \((x - 5)^{2}\).

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I simplified the terms of \(2 \sqrt{20}+4 \sqrt{75},\) and then I was able to add the like radicals.

Can a real number be both rational and irrational? Explain your answer.

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

Factor Completely. $$x^{4}-5 x^{2} y^{2}+4 y^{4}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every rational number is an integer.
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