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Chapter 0: Problem 52

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

Short Answer

Expert verified
The factorization of the perfect square trinomial \(x^{2}-10x+25\) is \((x-5)^{2}\).

Step by step solution

01

Identify the trinomial as a perfect square

The trinomial provided is \(x^{2}-10 x+25\). We can see that this matches the form \(a^{2}-2ab + b^{2}\). Thus, \( a = x, b = 5\). Therefore, it is indeed a perfect square.
02

Break down the trinomial into its elements

This means identifying the perfect squares that make up the trinomial. The first term \(a^{2} = x^{2}\) represents \(a\), the second term \(-2ab = -10x\) represents \(-2*x*5\), and the third term \(b^{2}= 25\) represents \(5^{2}\). Thus, \(a = x\) and \(b = 5\).
03

Factorize the trinomial

A perfect square trinomial can be factorized as \((a-b)^{2}\) or \((a+b)^{2}\). Here, as the trinomial pattern is \( x^{2}-2*x* 5+5^{2}\), and the middle term is negative, the formula to be used is \((a - b)^{2}\). Thus, the factorization would be \((x-5)^{2}\).

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Most popular questions from this chapter

When graphing the solutions of an inequality, what does a parenthesis signify? What does a square bracket signify?

Explain why \(|x|<-4\) has no solution.

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{1}{a^{3}-b^{3}} \cdot \frac{a c+a d-b c-b d}{1}\right)-\frac{c-d}{a^{2}+a b+b^{2}}$$

Will help you prepare for the material covered in the next section. If the width of a rectangle is represented by \(x\) and the length is represented by \(x+200\), write a simplified algebraic expression that models the rectangle's perimeter.

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