Textbook Question
Solve each equation or inequality.
Ch. 1 - Equations and Inequalities
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 2, Problem 63
Solve each equation using the quadratic formula. See Examples 5 and 6.
Verified step by step guidance1
First, expand the left side of the equation by using the distributive property: multiply each term in (4x - 1) by each term in (x + 2). This gives you \(4x \cdot x + 4x \cdot 2 - 1 \cdot x - 1 \cdot 2\).
Simplify the expression from step 1 to get a quadratic expression in standard form. Combine like terms to write the equation as \(ax^2 + bx + c = 4x\).
Next, move all terms to one side of the equation to set it equal to zero. Subtract \$4x\( from both sides to get \)ax^2 + bx + c - 4x = 0$, then combine like terms again.
Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation in standard form \(ax^2 + bx + c = 0\). These coefficients will be used in the quadratic formula.
Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) using the values of \(a\), \(b\), and \(c\) found in step 4. This will give you the solutions for \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. Understanding how to rewrite equations into this standard form is essential before applying methods like the quadratic formula.
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Introduction to Quadratic Equations
Expanding and Simplifying Expressions
To solve the given equation, you must first expand the product (4x - 1)(x + 2) and then simplify the resulting expression. This step helps in rearranging the equation into the standard quadratic form.
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05:20Expanding Radicals
Quadratic Formula
The quadratic formula x = [-b ± √(b² - 4ac)] / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It requires identifying coefficients a, b, and c correctly and calculating the discriminant to find real or complex roots.
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06:36Solving Quadratic Equations Using The Quadratic Formula
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