Solve each equation by completing the square. 3x2−12x+11=$$03x^2 -12x+11= 0$$

Ch. 1 - Equations and Inequalities

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 67

2장, 문제 67

Solve each equation by completing the square. $3 x^{2} - 12 x + 11 = 0$

검증된 단계별 안내

Start with the given quadratic equation: $3x^2 - 12x + 11 = 0$.

Divide the entire equation by 3 to make the coefficient of $x^2$ equal to 1: $x^2 - 4x + \frac{11}{3} = 0$.

Isolate the constant term on one side: $x^2 - 4x = -\frac{11}{3}$.

To complete the square, take half of the coefficient of $x$ (which is $-4$), square it, and add it to both sides. Half of $-4$ is $-2$, and $(-2)^2 = 4$. So add 4 to both sides: $x^2 - 4x + 4 = -\frac{11}{3} + 4$.

Rewrite the left side as a perfect square: $(x - 2)^2 = -\frac{11}{3} + \frac{12}{3}$, then simplify the right side.

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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 identifying coefficients and applying methods like completing the square correctly.

Isolating the Variable

Isolating the variable involves manipulating the equation to express the variable term alone on one side. In completing the square, this often means dividing through by the coefficient of x² and rearranging terms to prepare for forming a perfect square.

Solve each equation by completing the square. 3x2−12x+11=03x^2 -12x+11= 0