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

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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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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