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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 101
Chapter 2, Problem 101

Solve each equation in Exercises 83–108 by the method of your choice. x2=4x−7x^2 = 4x - 7

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1
Rewrite the given equation \(x^2 = 4x - 7\) by moving all terms to one side to set the equation equal to zero. This gives: \(x^2 - 4x + 7 = 0\).
Identify the coefficients in the quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -4\), and \(c = 7\).
Use the quadratic formula to solve for \(x\). The quadratic formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots. Substitute the values: \(\Delta = (-4)^2 - 4(1)(7)\).
Evaluate the square root of the discriminant and then substitute back into the quadratic formula to find the two possible values of \(x\).

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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. It typically has two solutions, which can be real or complex. Understanding how to recognize and manipulate quadratic equations is essential for solving them.
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Rearranging Equations to Standard Form

To solve a quadratic equation, it is important to rewrite it in standard form (ax² + bx + c = 0). This involves moving all terms to one side of the equation, which allows the use of various solving methods like factoring, completing the square, or the quadratic formula.
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Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Choosing the appropriate method depends on the equation's form and complexity, and each method leads to finding the roots or solutions of the equation.
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Related Practice
Textbook Question

Use interval notation to represent all values of x satisfying the given conditions. y = |2x - 5| + 1 and y > 9

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Textbook Question

Solve each equation in Exercises 83–108 by the method of your choice. x2 - 4x + 29 = 0

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Textbook Question

Solve each equation. 5 - 12x = 8 - 7x - [6 ÷ 3(2 + 53) + 5x]

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Textbook Question

In Exercises 101–106, solve each equation. ∣x−5∣=2\(\left\)|\(\sqrt{x}\)-5\(\right\)|=2

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Textbook Question

Exercises 100–102 will help you prepare for the material covered in the next section. Factor: x2−6x+9x^2 - 6x + 9

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Textbook Question

In Exercises 91–100, find all values of x satisfying the given conditions.y1=6(2xx−3)2,y2=5(2xx−3),andy1 exceeds y2 by 6.y_1 = 6 \(\left\)( \(\frac{2x}{x - 3}\) \(\right\))^2, \(\quad\) y_2 = 5 \(\left\)( \(\frac{2x}{x - 3}\) \(\right\)), \(\quad\) \(\text{and}\) \(\quad\) y_1 \(\text{ exceeds }\) y_2 \(\text{ by }\) 6.

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