Textbook Question
Solve each equation in Exercises 47–64 by completing the square.
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1. Equations & Inequalities
Choosing a Method to Solve Quadratics
1:48 minutesProblem 96
Textbook Question
Solve each equation in Exercises 83–108 by the method of your choice.
Verified step by step guidance1
Rewrite the quadratic equation in standard form: \(x^2 - 6x + 9 = 0\).
Identify the coefficients: \(a = 1\), \(b = -6\), and \(c = 9\).
Calculate the discriminant using the formula \(\Delta = b^2 - 4ac\) to determine the nature of the roots.
Apply the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\) to find the solutions for \(x\).
Simplify the expression under the square root and then simplify the entire expression to write the solutions in simplest form.
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1mKey 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. It typically has two solutions, which can be real or complex numbers. Understanding the structure of quadratic equations is essential for solving them effectively.
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Factoring Quadratic Expressions
Factoring involves rewriting a quadratic expression as a product of two binomials. This method is useful when the quadratic can be easily factored, allowing you to set each factor equal to zero to find the solutions. It simplifies solving quadratics without using formulas.
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Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides a universal method to solve any quadratic equation. It uses the coefficients a, b, and c to find the roots, including complex solutions when the discriminant (b² - 4ac) is negative.
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