Quadratic equations are a big part of algebra. Whether you’re in school or just brushing up on math, it’s important to know how to solve a quadratic equation. This guide breaks it down into simple steps using different methods. Don’t worry—we’ll go slow, use clear examples, and help you feel confident.
What Is a Quadratic Equation?
A quadratic equation is an equation that includes a variable raised to the power of two. The general form looks like this:
ax² + bx + c = 0
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a, b, and c are constants
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x is the variable
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a ≠ 0 (because otherwise, it wouldn’t be a quadratic)
Quadratics are common in real-life problems like area, speed, and profit.
Main Methods to Solve a Quadratic Equation
There are four main ways to solve a quadratic equation. Let’s look at each method one at a time.
1. Solving by Factoring
This is the easiest method—if the equation is factorable.
Steps:
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Write the equation in standard form (ax² + bx + c = 0)
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Factor the left side into two brackets
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Set each bracket equal to zero
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Solve for x
Example:
x² + 5x + 6 = 0
→ (x + 2)(x + 3) = 0
→ x + 2 = 0 → x = -2
→ x + 3 = 0 → x = -3
Solutions: x = -2 and x = -3
2. Solving Using the Quadratic Formula
If the equation can’t be factored, use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
This formula works for any quadratic equation.
Steps:
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Identify a, b, and c from the equation
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Plug the values into the formula
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Simplify the square root
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Solve both possible values for x
Example:
2x² + 3x – 2 = 0
→ a = 2, b = 3, c = -2
x = (-3 ± √(9 + 16)) / 4
x = (-3 ± √25) / 4
x = (-3 ± 5) / 4
→ x = (2)/4 = 0.5
→ x = (-8)/4 = -2
Solutions: x = 0.5 and x = -2
3. Solving by Completing the Square
This method rewrites the equation as a perfect square.
Steps:
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Move the constant to the other side
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Add a number to both sides to complete the square
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Factor the left side as a square
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Take the square root of both sides
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Solve for x
Example:
x² + 6x + 5 = 0
→ x² + 6x = -5
→ Add (6/2)² = 9 to both sides
→ x² + 6x + 9 = 4
→ (x + 3)² = 4
→ x + 3 = ±2
→ x = -1 or x = -5
Solutions: x = -1 and x = -5
4. Solving by Graphing
This is a visual method. You graph the equation and find the x-values where the graph touches the x-axis.
Steps:
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Rewrite the equation in y = ax² + bx + c form
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Graph it
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Look for the x-intercepts (where y = 0)
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These are your solutions
You can use graphing calculators or apps for this.
How to Know Which Method to Use
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If it factors easily → Use factoring
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If it doesn’t factor → Use the quadratic formula
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Want a clear step method → Complete the square
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Want to visualize → Graph it
Each method has its strengths. With practice, you’ll know which to use quickly.
What If the Equation Has No Real Solution?
Sometimes, the part under the square root (called the discriminant) is negative. That means the quadratic equation has no real solution, but it may have complex solutions.
Discriminant = b² – 4ac
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If it’s positive → 2 real solutions
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If it’s zero → 1 real solution
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If it’s negative → 2 complex solutions
Practice Problem
Try this one on your own:
x² – 4x – 5 = 0
Can you factor it?
(Hint: What two numbers multiply to -5 and add up to -4?)
Answer: (x – 5)(x + 1) = 0 → x = 5, x = -1
Final Thoughts
Now you know how to solve a quadratic equation using different methods. Start with factoring, then move to the formula or complete the square if needed. Don’t forget—practice makes perfect.
When in doubt, plug your solution back into the original equation to check your answer. Solving quadratics doesn’t have to be hard if you follow simple steps.
