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 given equation \(x^2 = 4x - 7\) by moving all terms to one side to set the equation equal to zero. This gives: \(x^2 - 4x + 7 = 0\).
2. Identify the coefficients in the quadratic equation \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -4\), and \(c = 7\).
3. Use the quadratic formula to solve for \(x\). The quadratic formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
4. Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots. Substitute the values: \(\Delta = (-4)^2 - 4(1)(7)\).
5. Evaluate the square root of the discriminant and then substitute back into the quadratic formula to find the two possible values of \(x\).

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. It typically has two solutions, which can be real or complex. Understanding how to recognize and manipulate quadratic equations is essential for solving them.

Rearranging Equations to Standard Form

To solve a quadratic equation, it is important to rewrite it in standard form (ax² + bx + c = 0). This involves moving all terms to one side of the equation, which allows the use of various solving methods like factoring, completing the square, or the quadratic formula.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Choosing the appropriate method depends on the equation's form and complexity, and each method leads to finding the roots or solutions of the equation.