Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 47

2장, 문제 47

Solve each equation. √(3x+7) = 3x+5

검증된 단계별 안내

Start with the given equation: \(\sqrt{3x + 7} = 3x + 5\).

To eliminate the square root, square both sides of the equation: \(\left(\sqrt{3x + 7}\right)^2 = (3x + 5)^2\).

Simplify both sides: \(3x + 7 = (3x + 5)^2\).

Expand the right side using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \(3x + 7 = 9x^2 + 30x + 25\).

Rearrange the equation to set it equal to zero: \(0 = 9x^2 + 30x + 25 - 3x - 7\), then combine like terms to get a quadratic equation.

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주요 개념

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Solving Radical Equations

Radical equations involve variables inside a root, often a square root. To solve them, isolate the radical expression and then eliminate the root by raising both sides to the appropriate power, typically squaring for square roots. This process can introduce extraneous solutions, so checking all solutions in the original equation is essential.

Domain Restrictions

When dealing with square roots, the expression inside the root must be non-negative to produce real numbers. This restriction limits the possible values of the variable, so determining the domain before solving helps avoid invalid solutions and simplifies the solving process.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. After finding potential solutions, substitute them back into the original equation to verify their validity and discard any extraneous solutions.

관련 실천

교과서 질문

Which equation has two real, distinct solutions? Do not actually solve.

A. (3x-4)² = -9 B. (4-7x)² = 0 C. (5x-9)(5x-9) = 0 D. (7x+4)² = 11

872

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교과서 질문

Height of a Projected Ball An astronaut on the moon throws a baseball upward. The astronaut is 6 ft, 6 in. tall, and the initial velocity of the ball is 30 ft per sec. The height s of the ball in feet is given by the equations=-2.7t²+30t+6.5,where t is the number of seconds after the ball was thrown. (a) After how many seconds is the ball 12 ft above the moon's surface? Round to the nearest hundredth. (b) How many seconds will it take for the ball to hit the moon's surface? Round to the nearest hundredth.

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교과서 질문

Solve each quadratic inequality. Give the solution set in interval notation. x(x-1)≤6

466

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교과서 질문

Solve each equation or inequality. | 3x + 1 | - 1 < 2

876

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교과서 질문

Solve each formula for the specified variable. Assume that the denominator is not 0 if variables appear in the denominator. s = 1/2gt², for g (distance traveled by a falling object)

867

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교과서 질문

Find each sum or difference. Write answers in standard form. (3+2i) + (9+3i)

768

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