How to Solve Quadratic Equations: A Complete Guide
Quadratic equations are fundamental in algebra and have numerous applications in science, engineering, and everyday problem-solving. This guide covers everything you need to know to master solving these equations.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the standard form:
ax² + bx + c = 0
Where:
a, b, and c are constants
a ≠ 0 (otherwise it would be a linear equation)
x is the variable we’re solving for
Four Methods to Solve Quadratic Equations
- Factoring Method
The factoring method works well when the quadratic expression can be easily factored.
Steps:
Ensure the equation is in standard form (ax² + bx + c = 0)
Factor the left side into the product of two binomials: (px + q)(rx + s)
Set each factor equal to zero and solve for x
Example:
Solve x² – 5x + 6 = 0
Solution:
Factor: (x – 2)(x – 3) = 0
Set each factor to zero:
x – 2 = 0, so x = 2
x – 3 = 0, so x = 3
The solutions are x = 2 and x = 3
- The Quadratic Formula
The quadratic formula works for ANY quadratic equation and is derived by completing the square on the standard form.
For ax² + bx + c = 0, the solutions are:
x = (-b ± √(b² – 4ac)) / 2a
Example:
Solve 2x² – 7x + 3 = 0
Solution:
a = 2, b = -7, c = 3
x = (7 ± √(49 – 24)) / 4
x = (7 ± √25) / 4
x = (7 ± 5) / 4
x = 3 or x = 1/2
- Completing the Square
This method transforms the quadratic equation into a perfect square trinomial plus a constant.
Steps:
Move the constant term to the right side
Divide by the coefficient of x² if necessary
Add and subtract (b/2a)² to complete the square
Rewrite as a squared binomial
Solve for x
Example:
Solve x² + 6x + 8 = 0
Solution:
Rearrange: x² + 6x = -8
Half the coefficient of x: 6/2 = 3
Square it: 3² = 9
Add and subtract 9: x² + 6x + 9 – 9 = -8
Rewrite: (x + 3)² = 1
Take square root: x + 3 = ±1
Solve: x = -3 ± 1, giving x = -2 or x = -4
- Using the Discriminant
The discriminant (b² – 4ac) helps determine the nature of the roots:
If b² – 4ac > 0: Two distinct real roots
If b² – 4ac = 0: One real root (repeated)
If b² – 4ac < 0: Two complex conjugate roots
Special Cases
- Perfect Square Trinomials
When a quadratic equation is in the form x² + 2kx + k² = 0, it can be rewritten as (x + k)² = 0, giving x = -k (repeated root).
- Difference of Squares
Equations in the form x² – d = 0 can be factored as (x – √d)(x + √d) = 0, giving x = ±√d.
Tips for Success
Always check your answers by substituting back into the original equation
Choose the appropriate method based on the equation’s form
For complex coefficients, the quadratic formula is usually the most reliable
Practice factoring to become more efficient at solving simpler quadratics
Real-World Applications
Quadratic equations are used in:
Physics (projectile motion)
Economics (profit optimization)
Engineering (design calculations)
Computer graphics (parabolic paths)
Finance (compound interest problems)
Master these techniques, and you’ll be able to tackle quadratic equations with confidence!