Solve each equation using completing the square. -3x² + 6x + 5 = 0

Verified step by step guidance

1. First, rewrite the equation to isolate the quadratic and linear terms on one side: \(-3x^2 + 6x = -5\).
2. Next, factor out the coefficient of \(x^2\) from the terms involving \(x\): \(-3(x^2 - 2x) = -5\).
3. Divide both sides by \(-3\) to simplify: \(x^2 - 2x = \frac{5}{3}\).
4. To complete the square, take half of the coefficient of \(x\) (which is \(-2\)), square it, and add it to both sides: half of \(-2\) is \(-1\), and \((-1)^2 = 1\), so add \(1\) to both sides to get \(x^2 - 2x + 1 = \frac{5}{3} + 1\).
5. Rewrite the left side as a perfect square: \((x - 1)^2 = \frac{5}{3} + 1\), then simplify the right side and solve for \(x\) by taking the square root of both sides.

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 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. To complete the square, the equation must be manipulated so that the coefficient of x² is 1, often requiring division of the entire equation by 'a' if 'a' is not 1.

Isolating the Variable

After forming a perfect square trinomial, the next step is to isolate the squared term and then take the square root of both sides. This yields two possible solutions, corresponding to the positive and negative roots, which are then simplified to find the values of x.