Quadratic Equations: Solving Mathematical Mysteries
by Yuvi K - December 16, 2023
Quadratic Equations: The Math Mystery
Math can be intimidating to students, especially when a quadratic equation is involved. However, quadratic equations are actually quite simple and can be solved if you understand the basic principles. In this article, we will explain what quadratic equations are, how to solve them, and provide a few examples of these equations in action.
What is a Quadratic Equation?
A quadratic equation is an equation that contains at least one term with an exponent of two. In algebra, the general form of a quadratic equation looks like ax2 + bx + c = 0, where a, b, and c are constants.
A quadratic equation can be solved either by factoring the equation or using the quadratic formula.
Factoring the Equation
Factoring is the process of breaking down an equation into its individual parts. The first step in factoring a quadratic equation is to rewrite the equation in the form ax2 + bx + c = 0, for example, 4x2 + 7x − 10 = 0.
Once the equation is in this form, the next step is to look for two numbers that when multiplied together equal the coefficient of the x2 term (a) and when added together equal the coefficient of the x term (b). In this example, the numbers are −2 and −5 since 4 × (−2) = (−5) + (−2) = −7.
To complete the factoring process, the equation must now be rewritten as (2x + 5)(2x -5) = 0. Now that the equation is partially factored, the final step is to set both factors equal to zero. In this case, 2x + 5 = 0 and 2x − 5 = 0. Solving each equation will result in x = −5/2, and x = 5/2.
Using the Quadratic Formula
The quadratic formula is another way of solving a quadratic equation. The quadratic formula is: x = (−b ± √(b2 − 4ac)) / 2a. To use the formula, the equation must first be written in the form ax2 + bx + c = 0.
Once the equation is in this form, the values for “a”, “b” and “c” must be determined. In the example equation 4x2 + 7x − 10 = 0, a is 4, b is 7, and c is −10.
Next, the quadratic formula is used to solve for x. When the equation is plugged into the quadratic formula, the answer for x is (7 ± √(49 + 40)) / 8. This simplifies to (7 ± 9) / 8, which equals 8/8 or x = 1, or -1/2 or x = −5/2.
Examples of Quadratic Equations
| Equation | Solution |
|---|---|
| 3x2 + 5x + 2 | x = (-5 ± √17) / 6, or x = -1 or x = 0.5 |
| 4x2 + 7x − 10 | x = (-7 ± √(49 + 40)) / 8, or x = 1 or x = -5/2 |
Conclusion
Quadratic equations can be intimidating at first, but, as you can see, they’re really not that difficult to solve. The key is understanding the basic principles of factoring and using the quadratic formula.
Once you understand these concepts, you’ll have no trouble solving any quadratic equations you come across in your math studies.
