Problem 14

Question

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

Step-by-Step Solution

Verified

Answer

The solutions are $x = -4$ and $x = 2$.

1Step 1 - 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 $c = -8$.

2Step 2 - 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 $(x + 4)(x - 2) = 0$.

3Step 3 - 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 $x - 2 = 0$.

4Step 4 - 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 $x = 2$.

Key Concepts

Quadratic EquationsFactoringSolving Equations Using the Zero-Factor Property

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’s 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’s how to apply it: