Quadratic Equation- Solutions and Examples
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The "2" in x² is what makes it quadratic — that exponent is the defining feature.
These equations pop up everywhere: physics, engineering, finance, computer graphics. If you're solving any problem involving parabolic motion or optimizing something, you're working with quadratics.
The Standard Form
Every quadratic equation looks like this:
ax² + bx + c = 0
Where:
- a = coefficient of x² (cannot be zero)
- b = coefficient of x
- c = constant term
Example: In 2x² + 5x - 3 = 0, you have a = 2, b = 5, c = -3.
How Many Solutions Does a Quadratic Have?
Most quadratics have two solutions. Sometimes these solutions are the same number (one repeated root). Sometimes they're not real numbers at all.
That's where the discriminant comes in — it tells you what you're dealing with before you even solve.
The Discriminant: Your First Step
The discriminant is the part under the square root in the quadratic formula: b² - 4ac
It tells you everything about the solutions:
- Positive (b² - 4ac > 0) → Two distinct real solutions
- Zero (b² - 4ac = 0) → One repeated real solution
- Negative (b² - 4ac < 0) → Two complex solutions (involving i)
Always check the discriminant first. It saves you time.
Methods for Solving Quadratic Equations
You have four main approaches. Each works, but some are faster depending on the equation.
1. Factoring
Factoring is the fastest method when it works. You're looking for two binomials that multiply together to give you the original equation.
Example: x² - 5x + 6 = 0
Find two numbers that multiply to 6 and add to -5. Those numbers are -2 and -3.
So: (x - 2)(x - 3) = 0
Solutions: x = 2 or x = 3
The problem? Many quadratics don't factor neatly, especially when roots are messy decimals or irrationals.
2. Quadratic Formula
This works for every quadratic equation. Memorize it:
x = (-b ± √(b² - 4ac)) / 2a
Plug in a, b, and c. Do the math. Done. No guessing, no hoping for nice numbers.
3. Completing the Square
This method rewrites the equation in the form (x - h)² = k. Useful when you need the vertex form of a parabola or when the quadratic formula feels too mechanical.
Steps:
- Move c to the other side
- Divide b by 2 and square it
- Add that value to both sides
- Factor the left side as a perfect square
- Solve for x
It takes more steps, but it builds understanding of why the quadratic formula works the way it does.
4. Graphical Method
Plot y = ax² + bx + c. The x-intercepts are your solutions. Useful for visualizing what's happening, but not practical for exact answers unless you have graphing software.
Comparing the Methods
| Method | Speed | Accuracy | Best When |
|---|---|---|---|
| Factoring | Fast | Exact | Numbers factor cleanly |
| Quadratic Formula | Moderate | Exact | Always works |
| Completing the Square | Slow | Exact | Need vertex form |
| Graphical | Varies | Approximate | Visual understanding |
Solved Examples
Example 1: Simple Factoring
Solve: x² + 7x + 12 = 0
Find two numbers that multiply to 12 and add to 7 → 3 and 4
(x + 3)(x + 4) = 0
Solutions: x = -3, x = -4
Example 2: Using the Quadratic Formula
Solve: 2x² + 4x - 6 = 0
Identify: a = 2, b = 4, c = -6
Discriminant: 4² - 4(2)(-6) = 16 + 48 = 64 (positive, so two real solutions)
x = (-4 ± √64) / 2(2)
x = (-4 ± 8) / 4
Solutions: x = 1 or x = -3
Example 3: One Repeated Root
Solve: x² - 6x + 9 = 0
Discriminant: (-6)² - 4(1)(9) = 36 - 36 = 0
x = (6 ± 0) / 2 = 3
Solution: x = 3 (double root)
Example 4: Complex Solutions
Solve: x² + 4x + 13 = 0
Discriminant: 16 - 52 = -36
x = (-4 ± √-36) / 2
√-36 = 6i
Solutions: x = -2 + 3i or x = -2 - 3i
How to Solve Any Quadratic Equation
Follow this process every time:
- Check the discriminant first. Know what kind of solutions you're dealing with before you start.
- Try factoring. If numbers are clean, this is fastest.
- Use the quadratic formula if factoring fails. It always works. Just be careful with signs.
- Verify your answers. Plug them back into the original equation. If it doesn't equal zero, you made an error.
Common Mistakes to Avoid
- Forgetting to set the equation equal to zero first. Move everything to one side before solving.
- Sign errors in the quadratic formula. The formula is (-b ± ...), not (b ± ...). The negative is part of it.
- Dividing by zero. Make sure a ≠ 0. If a = 0, it's not a quadratic.
- Not simplifying the discriminant. √64 = 8. Don't leave it as √64.
- Losing a negative sign when factoring. (x - 5)(x + 2) gives different results than (x + 5)(x - 2).
When You'll Actually Use This
Quadratics aren't just classroom exercises. Real applications:
- Projectile motion — finding when a ball hits the ground
- Area problems — maximizing space with limited fencing
- Business — break-even analysis and profit calculations
- Engineering — designing arches and suspension bridges
The math works the same whether you're calculating how long a rocket takes to return to earth or figuring out optimal pricing for a product.
The Bottom Line
Quadratic equations are straightforward once you know the options. Factor when it's easy. Use the formula when it's not. Check your discriminant at the start. Verify your answers at the end.
That's it. No magic, no special tricks. Just apply the method that fits the problem.