Textbook Question
In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 7 = 2x + 7
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 77
Solve each equation by the method of your choice.
Verified step by step guidance1
Start with the given equation: \( (x-3)^2 - 25 = 0 \).
Isolate the squared term by adding 25 to both sides: \( (x-3)^2 = 25 \).
Take the square root of both sides, remembering to consider both the positive and negative roots: \( x - 3 = \pm \sqrt{25} \).
Simplify the square root: \( x - 3 = \pm 5 \).
Solve for \(x\) by adding 3 to both sides for each case: \( x = 3 + 5 \) and \( x = 3 - 5 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Quadratic Equations
Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them involves finding values of x that satisfy the equation. Common methods include factoring, completing the square, and using the quadratic formula.
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Difference of Squares
The difference of squares is a special factoring technique where an expression of the form a² - b² can be factored into (a - b)(a + b). Recognizing this pattern simplifies solving equations like (x - 3)² - 25 = 0 by rewriting 25 as 5².
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Isolating the Variable
Isolating the variable means manipulating the equation to get x alone on one side. This often involves adding, subtracting, multiplying, dividing, or taking roots. It is a fundamental step in solving equations to find the exact values of the unknown.
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Related Practice
Textbook Question
In Exercises 59–94, solve each absolute value inequality. |3 - (2/3)x| > 5
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Textbook Question
Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).
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Textbook Question
In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation.
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Textbook Question
The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. |3x - 1| = |x + 5|
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Textbook Question
In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation.
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