Solve each equation using completing the square. 2x² + x = 10

검증된 단계별 안내

Start with the given quadratic equation: \(2x^2 + x = 10\).

Divide every term by 2 to make the coefficient of \(x^2\) equal to 1: \(x^2 + \frac{1}{2}x = 5\).

To complete the square, take half of the coefficient of \(x\), which is \(\frac{1}{2}\), divide it by 2 to get \(\frac{1}{4}\), then square it: \(\left(\frac{1}{4}\right)^2 = \frac{1}{16}\).

Add \(\frac{1}{16}\) to both sides of the equation to keep it balanced: \(x^2 + \frac{1}{2}x + \frac{1}{16} = 5 + \frac{1}{16}\).

Rewrite the left side as a perfect square trinomial: \(\left(x + \frac{1}{4}\right)^2 = 5 + \frac{1}{16}\), then solve for \(x\) by taking the square root of both sides and isolating \(x\).

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주요 개념

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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 both sides to create a binomial squared, making it easier to solve for the variable.

Quadratic Equation Standard Form

A quadratic equation is typically written in the form ax² + bx + c = 0. To use completing the square, the equation must first be rearranged into this standard form, isolating the quadratic and linear terms on one side and the constant on the other.

Isolating the Variable

Before completing the square, it is important to isolate the x-terms by dividing through by the coefficient of x² if it is not 1. This simplifies the process of forming a perfect square trinomial and helps in accurately solving for x.

Solve each equation using completing the square. 2x2 + x = 10