Quadratic Formula Examples- Solving Equations
What the Quadratic Formula Actually Is
The quadratic formula solves any equation in the form ax² + bx + c = 0. You plug in your numbers. You do the math. You get your answers. That's it.
Here's the formula:
x = (-b ± √(b² - 4ac)) / 2a
It works every time. Factoring fails. Completing the square gets messy. This formula handles everything.
When to Use the Quadratic Formula
Use it when:
- The equation doesn't factor cleanly
- You're dealing with ugly coefficients like 3x² + 7x - 2
- You need the exact answers, not decimal approximations
- Factoring would take longer than just plugging in
Skip it when the equation factors easily. If you see something like x² - 9 = 0, factor it. That's faster.
Reading the Formula: What Each Part Means
Before you solve anything, understand what you're looking at.
- a = coefficient of x²
- b = coefficient of x
- c = the constant term (no x attached)
The ± symbol means you'll get two answers — one using plus, one using minus. Every quadratic equation has at most two solutions.
Step-by-Step: How to Apply the Formula
Let's walk through the process before hitting examples.
- Make sure your equation is in ax² + bx + c = 0 form
- Identify a, b, and c from your equation
- Substitute them into the formula
- Calculate the discriminant (b² - 4ac) first
- Solve for x using both + and -
The discriminant tells you what kind of answers you'll get:
- Positive → two real solutions
- Zero → one repeated solution
- Negative → two complex solutions (involving i)
Quadratic Formula Examples: Easy to Hard
Example 1: x² - 5x + 6 = 0
This one factors easily (x-2)(x-3) = 0, but let's do it with the formula anyway.
Step 1: Identify a, b, c
a = 1, b = -5, c = 6
Step 2: Plug into the formula
x = -(-5) ± √((-5)² - 4(1)(6)) / 2(1)
x = 5 ± √(25 - 24) / 2
x = 5 ± √1 / 2
Step 3: Solve both
x = (5 + 1)/2 = 3
x = (5 - 1)/2 = 2
Answers: x = 2, x = 3. Same as factoring. The formula works.
Example 2: 2x² + 5x - 3 = 0
This one doesn't factor nicely. Let's go.
Step 1: a = 2, b = 5, c = -3
Step 2: Plug in
x = -5 ± √(5² - 4(2)(-3)) / 2(2)
x = -5 ± √(25 + 24) / 4
x = -5 ± √49 / 4
x = -5 ± 7 / 4
Step 3: Both solutions
x = (-5 + 7)/4 = 2/4 = 1/2
x = (-5 - 7)/4 = -12/4 = -3
Answers: x = 1/2, x = -3
Example 3: x² + 4x + 5 = 0
Let's see what happens with a negative discriminant.
a = 1, b = 4, c = 5
x = -4 ± √(16 - 20) / 2
x = -4 ± √(-4) / 2
The discriminant is negative. You get complex solutions:
x = -4 ± 2i / 2
x = -2 ± i
No real solutions. The parabola never touches the x-axis. That's fine — the formula still gives you the answer.
Example 4: 3x² - 7x = 2
Wait. This isn't in standard form yet. Move everything to one side first.
3x² - 7x - 2 = 0
Now identify: a = 3, b = -7, c = -2
x = -(-7) ± √((-7)² - 4(3)(-2)) / 2(3)
x = 7 ± √(49 + 24) / 6
x = 7 ± √73 / 6
Can't simplify √73. That's your answer. You can approximate it if needed:
x = (7 ± 8.544) / 6
x ≈ 15.544/6 ≈ 2.59 or x ≈ -1.544/6 ≈ -0.26
Quadratic Formula vs. Other Methods
Here's when each method makes sense:
| Method | Best When | Speed |
|---|---|---|
| Quadratic Formula | Ugly coefficients, doesn't factor, need exact answers | Slower but reliable |
| Factoring | Small, clean numbers, obvious factors | Fast if you spot it |
| Completing the Square | Vertex form needed, derivation required | Slowest, most steps |
| Graphing | Visual estimate, real-world context | Depends on tools |
The quadratic formula is your fallback. If you're stuck or the equation looks ugly, just use it.
Common Mistakes That Kill Your Answers
- Forgetting to set equation to zero. If you have x² + 3x = 10, you must write x² + 3x - 10 = 0 first.
- Sign errors with b. The formula has -b, not b. If b = -3, then -b = 3.
- Squaring b wrong. b² is always positive. (-7)² = 49.
- Dropping the ±. You always need both solutions unless the discriminant is zero.
- Forgetting to divide by 2a. The whole numerator goes over 2a, not just part of it.
Getting Started: Your Action Plan
Next time you see a quadratic equation:
- Check if it factors in 30 seconds. If yes, factor it. If no, proceed.
- Rewrite in ax² + bx + c = 0 form if needed.
- Write out a, b, c clearly.
- Calculate the discriminant (b² - 4ac) to know what you're dealing with.
- Plug into the formula. Don't rush the arithmetic.
- Simplify. Separate into two answers.
Practice with 10 problems. Start with simple ones like x² - 4x - 5 = 0. Work up to uglier ones like 6x² + x - 2 = 0. After a few sessions, you'll do this without thinking.