Quadratic equations are an essential part of Mathematics, and it is important to know how to solve them. There are different methods to solve quadratic equations, but in this article, we will focus on how to use the Quadratic Formula.
How to Solve a Quadratic Equation Using the Quadratic Formula
The Quadratic Formula is a universal formula that can be used to solve any quadratic equation in the form ax² + bx + c = 0. It works by providing the exact solutions to the equation no matter the values of a, b, and c.
Before we dive into how to solve a Quadratic equation, let us first look at some fundamental concepts of quadratic equations.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation in which the highest power of the variable is two. In other words, a quadratic equation must have an x² term. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. The values of a, b, and c determine the behavior of the quadratic equation.
For instance, if a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards. The vertex of the parabola is given by the point (-b/2a, f(-b/2a)), where f(x) is the quadratic function.
What is the Quadratic Formula?
The Quadratic Formula is a formula that provides the solutions of any quadratic equation in the form ax² + bx + c = 0. It is also known as the “Complete the Square” formula since its derivation involves completing the square of the quadratic expression.
The Quadratic Formula states that the solutions of the quadratic equation ax² + bx + c = 0 are given by:
x = (-b ± √(b² – 4ac)) / (2a)
This formula provides two values for x, which are called the roots of the equation. If the value under the square root sign is negative, then the equation has no real solutions.
Step-by-Step Guide on How to Use the Quadratic Formula
Now that we understand the fundamentals of quadratic equations and the Quadratic Formula, let us dive into how to use the formula. Here are the steps:
Step 1: Identify the Values of a, b, and c
The first step in using the Quadratic Formula is to identify the values of a, b, and c in the quadratic equation ax² + bx + c = 0. This information will be used to substitute the values in the formula.
Step 2: Substitute the Values in the Formula
Once you have identified the values of a, b, and c, you can substitute them in the Quadratic Formula. Recall that the formula is:
x = (-b ± √(b² – 4ac)) / (2a)
Substitute the values of a, b, and c in the formula, and simplify the expression. Since the formula provides two possible solutions, the expression can be split into two parts.
For instance, suppose you are given the equation 2x² + 5x – 3 = 0. You can identify that a = 2, b = 5, and c = -3. Then, substitute those values in the formula to get:
x = (-5 ± √(5² – 4(2)(-3))) / (2(2))
Simplify the expression:
x₁ = (-5 + √49) / 4 = -1
x₂ = (-5 – √49) / 4 = -3/2
Therefore, the roots of the quadratic equation 2x² + 5x – 3 = 0 are -1 and -3/2.
Step 3: Check Your Answers
Always check your answers to ensure they are correct. You can check your solutions by substituting them back into the original equation and verifying that the equation is satisfied.
Final Thoughts
The Quadratic Formula is an essential tool for solving quadratic equations. It provides a universal method of finding the roots of any quadratic equation in the standard form ax² + bx + c = 0.
However, it is crucial to understand the formula’s derivation and the underlying concepts of quadratic equations to use it effectively.
Practice using the Quadratic Formula on different types of quadratic equations to become proficient in solving them.