Use Choices A–D to answer each question. A. 3x2 - 17x - 6 = 0 B. (2x + 5)2 = 7 C. x2 + x = 12 D. (3x - 1)(x - 7) = 0 Which equation is set up for direct use of the square root property? Solve it.

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 10

2장, 문제 10

Use Choices A–D to answer each question. A. 3x² - 17x - 6 = 0 B. (2x + 5)² = 7 C. x² + x = 12 D. (3x - 1)(x - 7) = 0 Which equation is set up for direct use of the square root property? Solve it.

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Identify the equation that can be solved directly by applying the square root property. The square root property is used when an equation is in the form \(a^2 = k\), allowing you to take the square root of both sides.

Look at each choice to find an equation that can be rewritten as a perfect square equal to a constant. Choice B is \((2x + 5)^2 = 7\), which fits this form perfectly.

Apply the square root property to the equation \((2x + 5)^2 = 7\). Take the square root of both sides to get \(2x + 5 = \pm \sqrt{7}\).

Solve for \(x\) by isolating it. Subtract 5 from both sides: \(2x = -5 \pm \sqrt{7}\).

Finally, divide both sides by 2 to solve for \(x\): \(x = \frac{-5 \pm \sqrt{7}}{2}\).

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주요 개념

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Square Root Property

The square root property states that if x² = k, then x = ±√k. This property is used to solve equations where the variable is isolated and squared, allowing direct extraction of the square root to find solutions.

Identifying Equations Suitable for the Square Root Property

Equations suitable for the square root property are those that can be written in the form (expression)² = number. Recognizing such equations helps in applying the property directly without needing to factor or rearrange extensively.

Solving Quadratic Equations by Taking Square Roots

Once an equation is in the form (expression)² = k, solve by taking the square root of both sides, remembering to include both positive and negative roots. This method provides solutions quickly when applicable.