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

Verified step by step guidance

1. Rewrite the equation in standard quadratic form \(ax^2 + bx + c = 0\). Start by adding 7 to both sides to get \(x^2 - 6x + 7 = 0\).
2. Identify the coefficients: \(a = 1\), \(b = -6\), and \(c = 7\) from the equation \(x^2 - 6x + 7 = 0\).
3. Write down the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
4. Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(7)}}{2(1)}\).
5. Simplify inside the square root and the numerator step-by-step to prepare for solving for \(x\).

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 given 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 ax² + bx + c = 0. It uses the coefficients a, b, and c to find the roots, including real and complex solutions.

Discriminant and Nature of Roots

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If it is positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.