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Chapter 6: Problem 16

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$24 x^{2}+x-10=0$$

Short Answer

Expert verified
The solutions are \(x = \frac{5}{8}\) and \(x = -\frac{2}{3}\).

Step by step solution

01

Identify the Quadratic Equation

We're going to solve the quadratic equation \(24x^2 + x - 10 = 0\) by factoring. Our goal is to break it down into a product of binomials.
02

Factor the Quadratic Expression

We need to express \(24x^2 + x - 10\) as a product of two binomials. Start by finding the factors of the product \(24 \times -10 = -240\) that add up to 1 (the coefficient of \(x\)). These factors are 16 and -15. Rewrite \(x\) as \(16x - 15x\).
03

Apply Grouping Method

Rewrite \(24x^2 + x - 10\) as \(24x^2 + 16x - 15x - 10\). Group the terms into \((24x^2 + 16x) + (-15x - 10)\).
04

Factor by Grouping

Factor out common factors in each group: \(8x(3x + 2) - 5(3x + 2)\). Notice that \(3x + 2\) is a common factor.
05

Solve Factored Equation

Factor out the common binomial: \((8x - 5)(3x + 2) = 0\). Use the zero-product property to solve: \(8x - 5 = 0\) or \(3x + 2 = 0\).
06

Solve Each Binomial

For \(8x - 5 = 0\), solve to get \(x = \frac{5}{8}\). For \(3x + 2 = 0\), solve to get \(x = -\frac{2}{3}\).

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Key Concepts

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

Factoring
Factoring is a mathematical process where we express a polynomial as a product of simpler polynomials, called factors. In terms of quadratic equations, it involves rewriting the equation in a manner that simplifies solving it. For example, in the equation \(24x^2 + x - 10 = 0\), the aim is to split it into two binomials that when multiplied yield the quadratic expression.Steps Involved in Factoring Quadratics:
  • First, multiply the coefficient of \(x^2\) by the constant term to find a product (in this case, \(24 \times -10 = -240\)).
  • Next, find two numbers that multiply to \(-240\) and add to the linear coefficient, which is 1 (here, \(16\) and \(-15\)).
  • Rewrite the middle term using these two numbers, i.e., \(x\) becomes \(16x - 15x\).
  • Use grouping to rearrange the quadratic: \(24x^2 + 16x - 15x - 10\).
  • Factor by grouping to uncover the final factorized form, such as \((8x - 5)(3x + 2)\).
Zero-Product Property
The zero-product property is a key algebraic rule which states that if a product of two numbers is zero, at least one of the numbers must be zero. This concept is crucial for solving quadratic equations after factoring.Application:
  • Consider the factored form \((8x - 5)(3x + 2) = 0\).
  • Apply the zero-product property: if the product is zero, then \(8x - 5 = 0\) or \(3x + 2 = 0\).
  • Each of these equations can be solved individually to find the roots of the quadratic equation.
Using this property allows us to find the solutions neatly and confirms that the logic of factoring is correct.
Binomials
Binomials are algebraic expressions containing two terms, which can be separated by a plus or minus sign. In the process of factoring quadratic equations, binomials are indispensable.Understanding Binomials in Quadratic Factoring:
  • The goal in factoring is to express the quadratic equation as a product of binomials, such as \((8x - 5)\) and \((3x + 2)\).
  • These binomials can typically involve forms like \((ax + b)\), which are structured to simplify solving the equation.
  • The binomials reflect the possible roots of the equation when set to zero, like \(8x - 5 = 0\) and \(3x + 2 = 0\).
Through breaking down quadratics into binomials, we are simplifying complex expressions, making them more approachable for solving.
Solving Quadratic Equations
Solving quadratic equations is a fundamental task in algebra. It involves finding the values of \(x\) that make the equation true. Factoring, coupled with the zero-product property, is among the most straightforward methods.Procedure for Solving:
  • Begin with factoring the quadratic equation, such as \(24x^2 + x - 10 = 0\), into its binomial factors.
  • Apply the zero-product property which yields separate equations: \(8x - 5 = 0\) and \(3x + 2 = 0\).
  • Solve each equation independently to find \(x = \frac{5}{8}\) and \(x = -\frac{2}{3}\), which are the solutions to the original quadratic equation.
This method is efficient because it transforms complex expressions into simpler linear components, making calculations manageable.

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