Solve each equation. -2x² +11x = -21

Verified step by step guidance

1. Start by rewriting the equation to set it equal to zero: add 21 to both sides to get \(-2x^{2} + 11x + 21 = 0\).
2. Multiply the entire equation by -1 to make the leading coefficient positive: \$2x^{2} - 11x - 21 = 0$.
3. Identify the coefficients for the quadratic formula: \(a = 2\), \(b = -11\), and \(c = -21\).
4. Apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), substituting the values of \(a\), \(b\), and \(c\).
5. Simplify under the square root (the discriminant), then simplify the entire expression to find the two possible values of \(x\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. Solving such equations involves finding the values of x that satisfy the equation, which can be done by factoring, completing the square, or using the quadratic formula.

Rearranging Equations to Standard Form

Before solving, it is essential to rewrite the equation so that one side equals zero, typically in the form ax² + bx + c = 0. This allows the use of standard methods for solving quadratic equations and simplifies identifying coefficients a, b, and c.

Factoring and the Zero Product Property

Factoring involves expressing a quadratic as a product of two binomials. Once factored, the zero product property states that if the product of two factors is zero, then at least one factor must be zero. This principle helps find the solutions to the equation.