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Chapter 5: Problem 46

Factor. $$x^{2}+5 x-24$$

Short Answer

Expert verified
The factorized form of the given expression \(x^{2}+5 x-24\) is \((x+8)(x-3)\

Step by step solution

01

Identify the Coefficients

The given quadratic expression is \(x^{2}+5 x-24\). Here, the coefficient of \(x^{2}\) (a) is 1, the coefficient of x (b) is 5, and the constant term (c) is -24.
02

Find the pair of numbers

We need to find two numbers such that they add up to 5 (the coefficient of x) and multiply to -24 (the constant term). The numbers 8 and -3 fit these conditions because \(8*-3=-24\) and \(8+(-3)=5\).
03

Factorize

Use the pair of numbers found in the previous step to factorize the quadratic expression. Write \(x^{2}+8 x-3x-24\) instead of \(x^{2}+ 5x -24\) as \(8 * -3 = -24\) and \(8 + -3 = 5\). Then, factor by grouping. The expression \(x^{2}+8x-3x-24\) can be grouped as \((x^{2}+8x)+(-3x-24)\). Then, factor out common factors to yield \(x(x+8)-3(x+8)\). Interestingly, both terms now have a common binomial factor \((x+8)\)
04

Rewrite the Expression

Using the common factor, write the expression as \((x+8)(x-3)\). This is the factorized form of the given quadratic expression

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