Quadratic equations are at the heart of many algebraic problems. Whether you’re a student tackling math homework or someone brushing up on core concepts, knowing how to solve any quadratic equation with confidence is essential. A quadratic equation follows this standard form:
ax² + bx + c = 0
Where:
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a, b, and c are constants (with a ≠ 0),
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x is the variable you’re solving for.
Quadratic equations can have two real solutions, one real solution, or two complex solutions. Fortunately, there are multiple methods to solve them—each with its advantages.
In this guide, we’ll walk through all the key methods you can use to solve any quadratic equation, step by step.
Step 1: Identify the Form of the Equation
Before choosing your method, ensure the equation is written in the standard form:
ax² + bx + c = 0
If not, rearrange it by moving all terms to one side.
Example:
Given: x² = 6x – 5
Rewritten: x² – 6x + 5 = 0
Step 2: Choose the Best Method
You can solve quadratic equations using:
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Factoring (when factorable)
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Completing the square
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Quadratic formula
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Graphing (to visualize solutions)
We’ll explore each one below.
Method 1: Solving by Factoring
This is the simplest method but only works when the equation can be factored easily.
Example: x² – 5x + 6 = 0
Step 1: Find two numbers that multiply to c = 6 and add up to b = -5
Answer: -2 and -3
Step 2: Rewrite the equation
(x – 2)(x – 3) = 0
Step 3: Apply the Zero Product Property
x – 2 = 0 → x = 2
x – 3 = 0 → x = 3
Solutions: x = 2, x = 3
Method 2: Solving by Completing the Square
Use this method when factoring doesn’t work easily.
Example: x² + 6x + 5 = 0
Step 1: Move the constant
x² + 6x = -5
Step 2: Add the square of half the x-coefficient to both sides
(½ of 6 is 3 → 3² = 9)
x² + 6x + 9 = -5 + 9
(x + 3)² = 4
Step 3: Solve by taking the square root
x + 3 = ±2 → x = -3 ± 2
Step 4: Find the two values
x = -1 or x = -5
Method 3: Using the Quadratic Formula
This method always works, even if the equation can’t be factored.
Quadratic formula:
x = [-b ± √(b² – 4ac)] / (2a)
Example: 2x² + 4x – 6 = 0
a = 2, b = 4, c = -6
Step 1: Plug values into the formula
x = [-4 ± √(4² – 4(2)(-6))] / (2 × 2)
x = [-4 ± √(16 + 48)] / 4
x = [-4 ± √64] / 4
x = [-4 ± 8] / 4
Step 2: Simplify
x = (4) / 4 = 1
x = (-12) / 4 = -3
Solutions: x = 1, x = -3
Method 4: Solving by Graphing
Graph the function y = ax² + bx + c and find the x-intercepts, where y = 0.
Example: y = x² – 4x – 5
Graph this equation and look where the curve (parabola) crosses the x-axis.
You’ll see it crosses at x = 5 and x = -1, which are the solutions.
Note: This method gives a visual representation but may be approximate unless using a graphing calculator or tool like Desmos.
Step 3: Check Your Solutions
Always plug your solutions back into the original equation to verify their correctness.
Example: Original: x² – 5x + 6 = 0
Try x = 2 → (2)² – 5(2) + 6 = 0 ✅
Try x = 3 → (3)² – 5(3) + 6 = 0 ✅
Step 4: Consider the Discriminant
The discriminant is part of the quadratic formula:
D = b² – 4ac
It tells you the nature of the roots:
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D > 0: Two distinct real solutions
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D = 0: One real solution (repeated root)
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D < 0: Two complex (imaginary) solutions
This helps you decide if graphing or factoring is appropriate before you start.
Conclusion
No matter how complex a quadratic equation seems, it can always be solved using one of these four methods. Start by writing the equation in standard form, analyze the discriminant if necessary, and choose the most effective method—factoring, completing the square, the quadratic formula, or graphing.
Mastering all four methods will give you the flexibility to solve any quadratic equation with confidence and accuracy.
