Solve each equation in Exercises 83–108 by the method of your choice. x^2 = 4x - 7

Verified step by step guidance

1. Rewrite the equation in standard quadratic form: \(x^2 - 4x + 7 = 0\).
2. Identify the coefficients: \(a = 1\), \(b = -4\), \(c = 7\).
3. Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
4. Substitute the coefficients into the quadratic formula: \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}\).
5. Simplify the expression under the square root and solve for \(x\).

Key Concepts

Quadratic Equations

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. 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 effectively.

Factoring

Factoring involves rewriting an expression as a product of its factors. For quadratic equations, this often means expressing the equation in a form like (x - p)(x - q) = 0, where p and q are the roots of the equation. This method is particularly useful when the quadratic can be easily factored, allowing for straightforward solutions by setting each factor to zero.

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves manipulating the equation to express it in the form (x - p)^2 = q, which can then be solved by taking the square root of both sides. This technique is especially helpful when the quadratic does not factor easily.