Quadratic equations are a cornerstone of high school algebra, appearing in physics, economics, and geometry. A quadratic equation is any equation that can be rearranged into the standard form ax² + bx + c = 0.
In this guide, we will look at the two most popular and useful methods for solving quadratics: factorization (or factoring) and the quadratic formula. We will walk through detailed examples for each, showing you how to solve quadratic equations quickly and reliably.
Method 1: Solving Quadratic Equations by Factorization
Factoring is the fastest method to solve quadratic equations, but it only works if the quadratic expression can be written as a product of two linear binomials.
Steps for Factoring:
- Rearrange the equation into standard form: ax² + bx + c = 0.
- Find two numbers that multiply to ac (a × c) and add up to b.
- Rewrite the middle term and factor by grouping (or use shortcut factoring if a = 1).
- Set each factor equal to zero (using the Zero Product Property).
- Solve each resulting linear equation.
Example Problem:
Solve by factorization: x² - 5x + 6 = 0
Here, a = 1, b = -5, and c = 6.
Step 1: Find two numbers that multiply to 6 and add up to -5. Those numbers are -2 and -3.
Step 2: Factor the quadratic equation:
Step 3: Set each binomial factor to 0:
x - 3 = 0 => x = 3
The solutions are x = 2 and x = 3.
Method 2: Using the Quadratic Formula
What happens if a quadratic equation cannot be factored? (For example, x² + 4x - 7 = 0). In those cases, the Quadratic Formula is your ultimate tool. It works for every quadratic equation, whether it has real, rational, or complex solutions.
The quadratic formula is:
Example Problem:
Solve using the quadratic formula: 2x² + 5x - 3 = 0
Here, a = 2, b = 5, and c = -3.
Step 1: Plug the values into the formula:
Step 2: Simplify the discriminant (the term inside the square root):
x = [ -5 ± √49 ] / 4
Step 3: Take the square root of 49, which is 7:
Step 4: Separate into two cases (+ and -):
Case 2 (-): x = (-5 - 7) / 4 = -12 / 4 = -3
The solutions are x = 1/2 and x = -3.
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? Frequently Asked Questions
How do you solve quadratic equations by factoring?
To solve by factoring, rewrite the equation in standard form (ax² + bx + c = 0). Find two numbers that multiply to (a * c) and add up to b. Rewrite the equation with these factors, set each binomial factor to zero, and solve the resulting linear equations.
What is the quadratic formula?
The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a. It is derived from completing the square of the general quadratic equation and works on all quadratic equations.
What is the discriminant in a quadratic equation?
The discriminant is the term under the square root in the quadratic formula, b² - 4ac. If the discriminant is positive, there are two real solutions. If it is zero, there is one real solution. If it is negative, there are two complex (imaginary) solutions.
