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Chapter 8: Problem 5

Use the zero-factor property to solve each equation. \(x^{2}+3 x+2=0\)

Short Answer

Expert verified
The solutions are \(x = -1\) and \(x = -2\).

Step by step solution

01

- Factor the quadratic equation

First, factor the equation \(x^{2}+3 x+2=0\) into two binomial expressions. To do this, find two numbers that multiply to 2 and add up to 3. These numbers are 1 and 2. So, we can rewrite the equation as: \((x + 1)(x + 2) = 0\)
02

- Apply the Zero-Factor Property

According to the zero-factor property, if the product of two factors is zero, then at least one of the factors must be zero. Set each factor in the equation \((x + 1)(x + 2) = 0\) equal to zero and solve for \(x\):\(x + 1 = 0\)\(x + 2 = 0\)
03

- Solve for x

Solve each equation from the previous step:From \(x + 1 = 0\), we get \(x = -1\).From \(x + 2 = 0\), we get \(x = -2\).

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Key Concepts

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

zero-factor property
The zero-factor property is essential for solving quadratic equations. It states that if the product of two numbers is zero, then at least one of those numbers must be zero. This principle can be used to solve quadratic equations that have been factored.
For example, if we have an equation like \((x + 1)(x + 2) = 0\), we apply the zero-factor property to set each factor equal to zero. This gives us two simpler equations to solve:\
    \
  • \(x + 1 = 0\)
    \
  • \(x + 2 = 0\)\
\Solving these, we get the solutions \((-1)\) and \((-2)\).
This property makes solving quadratic equations straightforward when the equation is properly factored.
factoring quadratics
Factoring quadratics is a crucial step in solving a quadratic equation. It involves rewriting the quadratic equation as a product of two binomial expressions. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). To factor it, find two numbers that multiply to \(ac\) and add up to \(b\).
For instance, in the equation \(x^2 + 3x + 2 = 0\), we seek two numbers that multiply to 2 (the constant term) and add up to 3 (the coefficient of \(x\)). These numbers are 1 and 2. Thus, we can factor the equation as:
\((x + 1)(x + 2) = 0\)
This step converts the quadratic into two linear factors, making it easier to solve.
solving for roots
Solving for the roots of a quadratic equation means finding the values of \(x\) that make the equation true. After factoring the quadratic, we use the zero-factor property to solve for \(x\).
Taking the factored equation\((x + 1)(x + 2) = 0\), we set each factor equal to zero:
  • \(x + 1 = 0\)
  • \(x + 2 = 0\)

Solving these two simple equations, we get the roots:
  • \(x = -1\)
  • \(x = -2\)

These roots are the solutions of the original quadratic equation. The solutions, \(x = -1\) and \(x = -2\), are the x-values where the quadratic expression equals zero.
Finding the roots is important as it tells us the points where a quadratic graph intersects the x-axis.

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Most popular questions from this chapter

Solve each inequality, and graph the solution set. $$ \frac{6}{x-1}<1 $$

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