Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Intro to Quadratic Equations

College Algebra

1. Equations & Inequalities

Intro to Quadratic Equations

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1:33 minutes

Problem 22

Textbook Question

Solve each equation using the zero-factor property. See Example 1. 9x^2 - 12x + 4 = 0

Verified step by step guidance

First, recognize that the equation is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, \( a = 9 \), \( b = -12 \), and \( c = 4 \).

Next, try to factor the quadratic expression \( 9x^2 - 12x + 4 \). Look for two numbers that multiply to \( a \times c = 9 \times 4 = 36 \) and add to \( b = -12 \).

The numbers that satisfy these conditions are \(-6\) and \(-6\). Rewrite the middle term \(-12x\) using these numbers: \( 9x^2 - 6x - 6x + 4 \).

Group the terms to factor by grouping: \((9x^2 - 6x) + (-6x + 4)\).

Factor out the greatest common factor from each group: \(3x(3x - 2) - 2(3x - 2)\). Notice that \((3x - 2)\) is a common factor, so factor it out: \((3x - 2)(3x - 2) = 0\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Zero-Factor Property

The Zero-Factor Property states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle is essential for solving polynomial equations, as it allows us to set each factor equal to zero to find the solutions of the equation.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Understanding the structure of quadratic equations is crucial for applying the zero-factor property, as it often involves factoring the quadratic into two binomials.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. For quadratic equations, this often means expressing the equation in a form like (px + q)(rx + s) = 0. Mastery of factoring techniques is vital for effectively using the zero-factor property to solve equations.