Solve each quadratic equation using the quadratic formula. See Example 7. x² - 6x = -7

Verified step by step guidance

1. Rewrite the quadratic equation in standard form \(ax^2 + bx + c = 0\). For the given equation \(x^2 - 6x = -7\), add 7 to both sides to get \(x^2 - 6x + 7 = 0\).
2. Identify the coefficients \(a\), \(b\), and \(c\) from the standard form. Here, \(a = 1\), \(b = -6\), and \(c = 7\).
3. Recall the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula gives the solutions to any quadratic equation.
4. Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \times 1 \times 7}}{2 \times 1}\).
5. Simplify inside the square root and the numerator step-by-step to find the two possible values of \(x\).

Key Concepts

Quadratic Equation Standard Form

A quadratic equation is typically written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. To apply the quadratic formula, the equation must first be rearranged into this form by moving all terms to one side.

Quadratic Formula

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation in standard form. 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.