Solve each equation using completing the square. 3x2 - 9x + 7 = 0

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 48

2장, 문제 48

Solve each equation using completing the square. 3x² - 9x + 7 = 0

검증된 단계별 안내

Start with the given quadratic equation: \(3x^2 - 9x + 7 = 0\).

Divide every term by the coefficient of \(x^2\) (which is 3) to make the coefficient of \(x^2\) equal to 1: \(x^2 - 3x + \frac{7}{3} = 0\).

Move the constant term to the right side of the equation: \(x^2 - 3x = -\frac{7}{3}\).

To complete the square, take half of the coefficient of \(x\) (which is \(-3\)), square it, and add it to both sides. Half of \(-3\) is \(-\frac{3}{2}\), and its square is \(\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\): \(x^2 - 3x + \frac{9}{4} = -\frac{7}{3} + \frac{9}{4}\).

Rewrite the left side as a perfect square trinomial: \(\left(x - \frac{3}{2}\right)^2 = -\frac{7}{3} + \frac{9}{4}\). Then, simplify the right side by finding a common denominator and combining the fractions.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 creating a binomial squared expression on one side, making it easier to solve for the variable by taking square roots.

Quadratic Equation Standard Form

A quadratic equation is typically written in the form ax² + bx + c = 0, where a, b, and c are constants. Understanding this form is essential for applying methods like completing the square, as it helps identify coefficients and manipulate the equation correctly.

Isolating the Variable

Isolating the variable involves rearranging the equation so that the variable term is alone on one side. In completing the square, this step is crucial to set up the equation for creating a perfect square trinomial and eventually solving for the variable.