Solve each equation using completing the square. -2x2 + 4x + 3 = 0

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 45

2장, 문제 45

Solve each equation using completing the square. -2x² + 4x + 3 = 0

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First, rewrite the equation to isolate the quadratic term with a positive coefficient. Start with the equation \(-2x^2 + 4x + 3 = 0\). Divide every term by \(-2\) to get \(x^2 - 2x - \frac{3}{2} = 0\).

Next, move the constant term to the other side of the equation: \(x^2 - 2x = \frac{3}{2}\).

To complete the square, take half of the coefficient of \(x\), which is \(-2\), divide it by 2 to get \(-1\), and then square it to get \(1\). Add this value to both sides: \(x^2 - 2x + 1 = \frac{3}{2} + 1\).

Rewrite the left side as a perfect square trinomial: \((x - 1)^2 = \frac{3}{2} + 1\). Simplify the right side by expressing \(1\) as \(\frac{2}{2}\) and adding the fractions.

Finally, take the square root of both sides to solve for \(x\): \(x - 1 = \pm \sqrt{\text{right side}}\). Then isolate \(x\) by adding \(1\) to both sides.

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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. To complete the square, the equation must be manipulated into this form, often requiring dividing through by the coefficient of x² if it is not 1.

Isolating the Variable

After completing the square, the equation is rewritten so that the variable term is isolated on one side. This allows taking the square root of both sides, leading to two possible solutions due to the ± sign.