Solve each equation using the quadratic formula. See Examples 5 and 6. x^2 - 6x = -7

Verified step by step guidance

1. First, rewrite the equation in standard quadratic form, which is \( ax^2 + bx + c = 0 \). Start by adding 7 to both sides to get \( x^2 - 6x + 7 = 0 \).
2. Identify the coefficients \( a \), \( b \), and \( c \) from the equation \( x^2 - 6x + 7 = 0 \). Here, \( a = 1 \), \( b = -6 \), and \( c = 7 \).
3. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \).
4. Substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula: \( x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1} \).
5. Simplify the expression under the square root (the discriminant) and then simplify the entire expression to find the values of \( x \).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Quadratic Formula

The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a). This formula provides a systematic way to calculate the roots of the equation, regardless of whether the equation can be factored easily. Familiarity with this formula is crucial for solving quadratic equations.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, which can then be solved more easily. This technique involves manipulating the equation to express it in the form (x - p)² = q. Understanding this method is beneficial as it not only aids in solving quadratic equations but also provides insight into the properties of parabolas.