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Chapter 8: Problem 8

Use the zero-factor property to solve each equation. \(2 x^{2}+x-6=0\)

Short Answer

Expert verified
The solutions are: x = 3/2 and x = -2.

Step by step solution

01

Write the equation in standard form

The given equation is already in standard form: 2x^2 + x - 6 = 0
02

Factor the quadratic equation

Factor the quadratic equation into two binomials. 2x^2 + x - 6 = (2x-3)(x+2) = 0
03

Apply the zero-factor property

Using the zero-factor property, set each factor equal to zero: (2x-3) = 0 (x+2) = 0
04

Solve each equation

Solve the equations obtained in Step 3 for x: For (2x-3)=0, solve for x: 2x = 3 x = 3/2 For (x+2)=0, solve for x: 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 a fundamental principle in algebra used to solve quadratic equations. This property states that if the product of two factors is zero, at least one of the factors must be zero. In other words, if (AB = 0), then at least one of the following must hold true: (A = 0) or (B = 0).
Let's see how this works with an example, using the equation from above:
  • The equation is factored into ( (2x-3)(x+2) = 0 ).
  • We then set each factor equal to zero: ( 2x-3=0 ) and ( x+2=0 ).
  • We solve each of these equations separately to find the solutions for x.
This simple yet powerful property is a key tool for solving quadratic equations efficiently.
Factoring Quadratic Equations
Factoring is a method used to break down a quadratic equation into simpler binomial expressions. For quadratic equations in the form (Ax^2 + Bx + C = 0), the goal is to find two binomials that multiply to give the original quadratic expression.
Here's a step-by-step guide on factoring the quadratic equation (2x^2 + x - 6 = 0):
  • Identify a pair of numbers that multiply to (a * c), where (a) is the coefficient of (x^2) and (c) is the constant term.
  • In this case, (a * c = 2 * (-6) = -12), and the numbers should also sum to (b), the coefficient of (x), which is 1.
  • The pair of numbers that work here are 3 and -4, since (3*-4 = -12) and (3 - 4 = 1).
  • Rewrite the middle term using these numbers: (2x^2 + 3x - 2x - 6 = 0).
  • Factor by grouping: ((2x^2 + 3x) - (2x + 6)) becomes (x(2x + 3) - 2(2x + 3) = 0).
  • Finally, factor out the common binomial: ((2x-3)(x+2) = 0).
The equation is now factored and ready for the zero-factor property to be applied.
Standard Form
Quadratic equations should be written in standard form to make them easier to solve. The standard form of a quadratic equation is (ax^2 + bx + c = 0),
where (a), (b), and (c) are constants, and (x) represents the variable.

Writing an equation in standard form is crucial because:
  • It allows you to identify the values of (a), (b), and (c) easily.
  • It makes factoring the equation more straightforward.
  • It helps in applying the quadratic formula if necessary.
In our example (2x^2 + x - 6 = 0), we can clearly see that (a = 2), (b = 1 ) , and (c = -6). This standard form setup makes the next steps of solving, whether by factoring, completing the square, or using the quadratic formula, much clearer and more methodical.

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Most popular questions from this chapter

Solve each problem. When appropriate, round answers to the nearest tenth. Manuel is planting a vegetable garden in the shape of a right triangle. The longer leg is \(3 \mathrm{ft}\) longer than the shorter leg, and the hypotenuse is \(3 \mathrm{ft}\) longer than the longer leg. Find the lengths of the three sides of the garden.

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ 9 x^{2}-12 x-1=0 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ (2 x-1)^{2}=x+2 $$

Use the zero-factor property to solve each equation. \(5 x^{2}=11 x-2\)

Solve each equation. Check the solutions. $$\frac{-12}{x}=x+8$$
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