Domanda del libro di testo
Solve each equation in Exercises 41–60 by making an appropriate substitution.
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 2, Problema 53
Solve each equation in Exercises 47–64 by completing the square.
Guida verificata passo dopo passo1
Start with the given quadratic equation: \(x^2 + 4x + 1 = 0\).
Move the constant term to the right side to isolate the \(x\) terms: \(x^2 + 4x = -1\).
To complete the square, take half of the coefficient of \(x\), which is 4, divide by 2 to get 2, then square it to get \(2^2 = 4\).
Add this square (4) to both sides of the equation to maintain equality: \(x^2 + 4x + 4 = -1 + 4\).
Rewrite the left side as a perfect square trinomial: \((x + 2)^2 = 3\). From here, you can proceed to solve for \(x\) by taking the square root of both sides.
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Completing the Square
Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves adding and subtracting a specific value to create a binomial squared, making it easier to solve for the variable.
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Solving Quadratic Equations by Completing the Square
Quadratic Equations
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Understanding its structure is essential for applying methods like completing the square, factoring, or using the quadratic formula to find the roots.
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Solving Equations Using Square Roots
Once a quadratic equation is written as a perfect square equal to a constant, you solve for the variable by taking the square root of both sides. This step introduces both positive and negative roots, which are critical for finding all solutions.
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06:12Solving Quadratic Equations by the Square Root Property
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