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

The Quadratic Formula

College Algebra

1. Equations & Inequalities

The Quadratic Formula

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1:48 minutes

Problem 96

Textbook Question

Solve each equation in Exercises 83–108 by the method of your choice.9 - 6x + x^2 = 0

Verified step by step guidance

Rewrite the equation in standard quadratic form: \(x^2 - 6x + 9 = 0\).

Recognize that this is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -6\), and \(c = 9\).

Notice that the quadratic can be factored as a perfect square: \((x - 3)^2 = 0\).

Set the factored expression equal to zero: \((x - 3)^2 = 0\).

Solve for \(x\) by taking the square root of both sides, leading to \(x - 3 = 0\), and then solve 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 polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. In the given equation, 9 - 6x + x^2 = 0, it can be rearranged to the standard form to identify the coefficients.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. For quadratic equations, this often involves finding two binomials that multiply to give the quadratic. In the context of the equation provided, factoring can simplify the process of finding the roots by setting each factor equal to zero.

Roots of an Equation

The roots of an equation are the values of the variable that satisfy the equation, making it true. For quadratic equations, the roots can be real or complex numbers, and they can be found using methods such as factoring, the quadratic formula, or graphing. Understanding how to find and interpret these roots is essential for solving the given equation effectively.