Solve each equation. -2x² +11x = -21

Verified step by step guidance

1. First, rewrite the equation in standard quadratic form: \(-2x^2 + 11x + 21 = 0\).
2. Next, identify the coefficients: \(a = -2\), \(b = 11\), and \(c = 21\).
3. Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
4. Substitute the coefficients into the quadratic formula: \(x = \frac{-11 \pm \sqrt{11^2 - 4(-2)(21)}}{2(-2)}\).
5. Simplify under the square root and solve for \(x\) using the quadratic formula.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. These equations can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the standard form and properties of quadratic equations is essential for solving them.

Factoring

Factoring involves rewriting an expression as a product of its factors. For quadratic equations, this often means expressing the equation in a form that can be set to zero, allowing for the identification of roots. Mastery of factoring techniques, such as finding common factors or using the difference of squares, is crucial for simplifying and solving quadratic equations.

The Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. It is particularly useful when the equation cannot be easily factored. Understanding how to apply the quadratic formula, including calculating the discriminant (b² - 4ac), helps determine the nature and number of solutions for the equation.