The Quadratic Formula
Learn to solve any quadratic equation using the quadratic formula.
Definition
- , , and are the coefficients from the equation
- (otherwise it's not quadratic)
- The symbol means there are usually two solutions
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Worked Examples
Solve using the quadratic formula.
Identify a, b, and c
Comparing to : , , →
Calculate the discriminant
→
Apply the formula
→
Find both solutions
→ or
Answer: or
Common Mistakes
Forgetting to make b negative when it's already negative
Why it's wrong: If , then , not . The formula says , so you negate whatever b is.
Correct: Always write first, then substitute: if , write
Calculating incorrectly when b is negative
Why it's wrong: Students sometimes write instead of
Correct: The entire coefficient is squared:
Dividing only part of the numerator by 2a
Why it's wrong: Writing instead of
Correct: The entire numerator is divided by
Stopping at one solution
Why it's wrong: The means plus OR minus, giving two potential solutions
Correct: Always calculate both: and
Interactive Visual
Linear Function Explorer
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
Equation Solver
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Follow along as we solve the equation step by step.
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y = 2x + 1
m=2, b=1
Expression Calculator
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Practice Problems
15 problemsIn the equation , what are the values of , , and ?
Why It Matters
- Universal: It works for ANY quadratic equation, even when factoring doesn't
- Predictable: Follow the same steps every time
- Real-world applications: Physics (projectile motion), engineering (parabolic structures), finance (profit optimization)
Real World Applications
Projectile Motion
When you throw a ball, its height follows a quadratic equation. The formula helps find when it lands.
Example:
A ball thrown upward has height meters after seconds. To find when it hits the ground (), solve using the formula.
A rocket's height is modeled by meters.
When does the rocket return to the ground?
Step 1: Write the mathematical expression
Set and identify the coefficients:
Business Profit Optimization
Companies use quadratic equations to model profit and find break-even points.
Example:
A company's profit is dollars, where is units sold. To find break-even points (where ), use the quadratic formula.
A shop's weekly profit is dollars for selling items.
At what sales levels does the shop break even?
Step 1: Write the mathematical expression
Solve :
Key Takeaways
- 1The quadratic formula is
- 2It works for any equation in the form
- 3The discriminant tells you how many solutions exist
- 4If : two distinct real solutions; if : one repeated solution; if : no real solutions
- 5Always check your answers by substituting back into the original equation
Frequently Asked Questions
Glossary
- Quadratic equation
- An equation of the form where
- Discriminant
- The expression that determines the number and type of solutions
- Coefficient
- The numerical factor of a term (e.g., in , the coefficient is 3)
- Root/Solution
- A value of that makes the equation true (where the parabola crosses the x-axis)
Formula Card
Quadratic Formula
Solves any equation $ax^2 + bx + c = 0$
Discriminant
Determines number and type of solutions
Two solutions
Discriminant positive: two distinct real roots
One solution
Discriminant zero: one repeated root
No real solutions
Discriminant negative: no real roots