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Chapter 1: Problem 14

Solve each equation by the zero-factor property. $$x^{2}+2 x-8=0$$

Short Answer

Expert verified
The solutions are \(x = -4\) and \(x = 2\).

Step by step solution

01

- Identify the quadratic equation

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 2\), and \(c = -8\).
02

- Factor the quadratic equation

To solve for \(x\), we will factor the quadratic equation \(x^2 + 2x - 8\). Look for two numbers that multiply to \(-8\) (the constant term) and add to \(2\) (the coefficient of \(x\)). The numbers \(4\) and \(-2\) work because \(4 \times -2 = -8\) and \(4 + (-2) = 2\). Therefore, the equation can be factored as \((x + 4)(x - 2) = 0\).
03

- Apply the zero-factor property

The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero. Thus, we set each factor equal to zero and solve for \(x\): \(x + 4 = 0\) or \(x - 2 = 0\).
04

- Solve each equation

Solving the two equations from Step 3: \(x + 4 = 0\) gives \(x = -4\) and \(x - 2 = 0\) gives \(x = 2\). Thus, the solutions to the equation \(x^2 + 2x - 8 = 0\) are \(x = -4\) and \(x = 2\).

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Key Concepts

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

Quadratic Equations
A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The highest exponent of the variable \(x\) is 2.
Here are some features:
  • It looks like a parabola when graphed.
  • It has two solutions (real or complex).
  • The term with \(x^2\) is the quadratic term.
  • The term with \(x\) is the linear term.
  • The constant term (\(c\)) is often referred to as the free term.
To solve a quadratic equation, you need to find the values of \(x\) that make the equation true. In our example, the quadratic equation to solve is \(x^2 + 2x - 8 = 0\).
Factoring
Factoring is a method used to break down a polynomial into simpler components, making it easier to solve. For quadratic equations, we aim to express the equation \(ax^2 + bx + c\) as the product of two binomials. Here鈥檚 how:
  • Find two numbers that multiply to \(a \times c\) (the product of the quadratic coefficient and the constant term).
  • These two numbers should also add up to the linear coefficient (\(b\)).
  • Write the quadratic equation in its factored form.
For the equation \(x^2 + 2x - 8 = 0\), we look for two numbers that multiply to -8 and add up to 2. The numbers 4 and -2 fit, allowing us to factor the equation as \((x + 4)(x - 2) = 0\).
Solving Equations Using the Zero-Factor Property
The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero. This is key in solving factored equations. Here鈥檚 how to apply it:
  • Set each factor equal to zero separately.
  • Solve each resulting equation for \(x\).
Applying this to \((x + 4)(x - 2) = 0\), we get:
  • \(x + 4 = 0 \Rightarrow x = -4\)
  • \(x - 2 = 0 \Rightarrow x = 2\)
Therefore, the solutions to the quadratic equation \(x^2 + 2x - 8 = 0\) are \(x = -4\) and \(x = 2\).

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