Solve each equation using completing the square. 3x² + 2x = 5

Verified step by step guidance

1. Start with the given quadratic equation: \$3x^2 + 2x = 5$.
2. Move the constant term to the left side to isolate the quadratic and linear terms: \$3x^2 + 2x - 5 = 0$.
3. Divide the entire equation by the coefficient of \(x^2\) (which is 3) to make the coefficient of \(x^2\) equal to 1: \(x^2 + \frac{2}{3}x = \frac{5}{3}\).
4. To complete the square, take half of the coefficient of \(x\), which is \(\frac{2}{3}\), divide it by 2 to get \(\frac{1}{3}\), then square it to get \(\left(\frac{1}{3}\right)^2 = \frac{1}{9}\). Add this value to both sides of the equation: \(x^2 + \frac{2}{3}x + \frac{1}{9} = \frac{5}{3} + \frac{1}{9}\).
5. Rewrite the left side as a perfect square trinomial: \(\left(x + \frac{1}{3}\right)^2 = \frac{5}{3} + \frac{1}{9}\). Then simplify the right side by finding a common denominator and adding the fractions.

Key Concepts

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 form 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 complete the square, the equation must first be rearranged so that the quadratic and linear terms are on one side and the constant on the other, often requiring division by the leading coefficient if it is not 1.

Solving Quadratic Equations

After completing the square, the equation can be solved by taking the square root of both sides, remembering to consider both positive and negative roots. This step isolates the variable and leads to the solution(s) of the quadratic equation.