Solve each equation in Exercises 47–64 by completing the square. $2 x^{2} - 7 x + 3 = 0$

검증된 단계별 안내

Start with the given quadratic equation: $2x^2 - 7x + 3 = 0$.

Divide the entire equation by the coefficient of $x^2$, which is 2, to make the coefficient of $x^2$ equal to 1: $x^2 - \frac{7}{2}x + \frac{3}{2} = 0$.

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

To complete the square, take half of the coefficient of $x$, which is $-\frac{7}{2}$, divide it by 2 to get $-\frac{7}{4}$, then square it to get $\left(-\frac{7}{4}\right)^2 = \frac{49}{16}$. Add this value to both sides of the equation: $x^2 - \frac{7}{2}x + \frac{49}{16} = -\frac{3}{2} + \frac{49}{16}$.

Rewrite the left side as a perfect square trinomial: $\left(x - \frac{7}{4}\right)^2 = -\frac{3}{2} + \frac{49}{16}$. Then simplify the right side by finding a common denominator and combining the terms.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

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 identifying coefficients and applying methods like completing the square 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 often includes dividing by the coefficient of x² and moving the constant term to the other side to prepare for forming a perfect square.

Solve each equation in Exercises 47–64 by completing the square. 2x2−7x+3=02x^2 - 7x + 3 = 0