Solve each equation using the quadratic formula. -4x² = -12x + 11

Verified step by step guidance

1. First, rewrite the equation in standard quadratic form, which is \(ax^2 + bx + c = 0\). Start with the given equation: \(-4x^2 = -12x + 11\). Move all terms to one side by adding \$12x\( and subtracting \(11\) from both sides to get \)-4x^2 + 12x - 11 = 0$.
2. Identify the coefficients \(a\), \(b\), and \(c\) from the standard form equation. Here, \(a = -4\), \(b = 12\), and \(c = -11\).
3. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula gives the solutions to any quadratic equation.
4. Calculate the discriminant, which is the expression under the square root: \(b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into this expression to find the discriminant.
5. Substitute the values of \(a\), \(b\), and the discriminant into the quadratic formula. Then simplify the expression step-by-step to find the two possible values of \(x\).

Key Concepts

Quadratic Equation Standard Form

A quadratic equation must be written in the standard form ax² + bx + c = 0 before applying the quadratic formula. This involves rearranging all terms to one side of the equation so that the other side equals zero, allowing identification of coefficients a, b, and c.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots, including real and complex solutions, depending on the discriminant.

Discriminant and Nature of Roots

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots. This helps anticipate the type of solutions before solving.