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Chapter 11: Problem 9

Solve using the zero-factor property. $$ x^{2}-x-56=0 $$

Short Answer

Expert verified
The solutions are \(x = 8\) and \(x = -7\).

Step by step solution

01

- Set the Equation to Zero

Begin by ensuring the quadratic equation is set equal to zero, which in this case it already is: \[x^{2} - x - 56 = 0\]
02

- Factor the Quadratic Equation

Factor the left-hand side of the equation. Look for two numbers that multiply to -56 and add up to -1. These numbers are -8 and 7. Therefore, the equation can be factored as:\[(x - 8)(x + 7) = 0\]
03

- Apply the Zero-Factor Property

According to the zero-factor property, if a product of factors equals zero, at least one of the factors must be zero. Set each factor equal to zero:\[x - 8 = 0\]\[x + 7 = 0\]
04

- Solve Each Equation

Solve for \(x\) in each equation separately:\[x - 8 = 0\]\[x = 8\]\[x + 7 = 0\]\[x = -7\]
05

- State the Solution

The solutions to the equation are \(x = 8\) and \(x = -7\).

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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 an essential concept when solving quadratic equations. It states that if the product of two factors is zero, then at least one of the factors must be zero.
For example, if you have an equation like \((a \times b = 0)\), then either \(a = 0\) or \(b = 0\). This property helps us break down complex equations into simpler forms.
Remember, this only applies when the equation is set to zero. Therefore, the first step in solving a quadratic equation using this method is to ensure the equation equals zero. Then, factorize the quadratic expression to find the roots.
factoring quadratics
Factoring quadratics involves rewriting a quadratic equation into a product of two binomial expressions. This process makes it easier to find the roots of the equation.
Let's take the quadratic equation from our problem: \((x^2 - x - 56 = 0)\).
To factorize this, look for two numbers that multiply to -56 and add to -1. These numbers are -8 and 7.
Then, the equation becomes: \((x - 8)(x + 7) = 0)\).
When you've factorized the quadratic completely, you're ready to apply the zero-factor property.
solving equations
Once you have the quadratic equation factored and set to zero, solving the equation becomes straightforward. Using the zero-factor property:
  • Set each factor equal to zero: \(x - 8 = 0\) and \(x + 7 = 0\)
  • Then, solve each simple equation to find the values of x: \(x = 8\) and \(x = -7\)

So, the solutions to the quadratic equation \((x^2 - x - 56 = 0)\) are \(x = 8\) and \(x = -7\).
Always remember to check your solutions by plugging them back into the original equation to ensure they satisfy the equation. This verification step helps confirm your answers are correct.

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Most popular questions from this chapter

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