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Chapter 10: Problem 60

For Exercises 7 to \(75,\) factor. $$x^{2}-4 x-96$$

Short Answer

Expert verified
The factored form of the quadratic expression \(x^{2}-4x-96\) is \((x-12)(x+8)\).

Step by step solution

01

Identifying the coefficients

In this quadratic equation, identify the coefficients as follows: the coefficient 'a' is 1, 'b' is -4, and 'c' is -96.
02

Find the factors

Look for two numbers that multiply to give the product of 'a' and 'c' (ac = 1*(-96) = -96) and add to give 'b' (-4). The numbers are 8 and -12 since 8*(-12) = -96 and 8 + (-12) = -4.
03

Write the factored form of the quadratic expression

The factored form of the quadratic expression \(x^{2}-4x-96\) is \((x-12)(x+8)\) as these satisfy the conditions identified in step 2. Hence, the quadratic equation in factored form is \((x-12)(x+8)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Expression
A quadratic expression is a type of polynomial that has a degree of two as its highest exponent. It typically looks like this: \(ax^2 + bx + c\), where:
  • 'a' is the coefficient of \(x^2\)
  • 'b' is the coefficient of \(x\)
  • 'c' is the constant term

In our example, the quadratic expression is \(x^2 - 4x - 96\). The fact that it's a quadratic expression means that it creates a parabola when graphed, which is a symmetrical, U-shaped curve. Quadratic expressions are fundamental in algebra, and learning how to manipulate them is a valuable skill. Understanding the structure of these expressions allows us to apply various techniques to solve or simplify them.
Coefficient Identification
Identifying coefficients is a critical step in working with quadratic equations. In the given expression \(x^2 - 4x - 96\), we need to spot the values of 'a', 'b', and 'c'. Here, the coefficients are as follows:
  • 'a' equals 1 since there is no number explicitly written before \(x^2\), which means it鈥檚 1 by default.
  • 'b' is -4, the coefficient of \(x\).
  • 'c' is -96, the constant term.

This identification is essential because these coefficients help us determine factors and subsequently solve the equation. They are the keys to unlocking the factored form of the quadratic expression.
Factored Form
The factored form of a quadratic expression is important because it simplifies solving the equation. To find the factored form, we look for two numbers that multiply to give the product of 'a' and 'c' (in our example, the product is -96), and add up to 'b' (-4 in this case).
For our expression \(x^2 - 4x - 96\), the numbers 8 and -12 fulfill these conditions since 8 multiplied by -12 equals -96, and 8 plus -12 equals -4.
Thus, the original quadratic expression can be rewritten in factored form as \((x-12)(x+8)\). This format will make solving the quadratic equation much easier.
Solving Quadratic Equations
Once we have the factored form of the quadratic equation, solving it becomes straightforward. Given the factored form \((x-12)(x+8)\), we can solve the equation by setting each factor equal to zero.
This gives us two separate equations:
  • \(x - 12 = 0\)
  • \(x + 8 = 0\)

Solving these equations provides the solutions \(x = 12\) and \(x = -8\). This process demonstrates how factoring a quadratic expression not only simplifies the expression but also helps us find its roots efficiently. Mastery of this technique is crucial because it appears frequently in algebraic problems.

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