Introduction to the Quadratic Formula
A quadratic equation takes the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. While some quadratics can be solved by factoring, the quadratic formula works for every quadratic — even those that don't factor neatly into integers.
The formula is derived from completing the square on the general quadratic, and it's worth memorizing. Once you know it, you have a universal tool for solving any quadratic equation you encounter.
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
This formula gives the solutions to ax² + bx + c = 0. The ± symbol means you get two answers: one using the plus sign and one using the minus sign.
Step-by-Step Application:
- Write the equation in standard form: ax² + bx + c = 0
- Identify the values of a, b, and c
- Compute the discriminant: b² − 4ac
- Substitute into the formula and simplify
- Check your solutions in the original equation
The Discriminant
The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many solutions to expect before you even compute them.
| Discriminant | Number of Solutions | Type of Solutions | Graph Behavior |
|---|---|---|---|
| Positive (b² − 4ac > 0) | Two distinct solutions | Real (rational or irrational) | Crosses x-axis twice |
| Zero (b² − 4ac = 0) | One solution (repeated) | Real, single root | Touch x-axis at vertex |
| Negative (b² − 4ac < 0) | No real solutions | Complex (imaginary) | Does not cross x-axis |
Worked Examples
Example 1: Two Distinct Solutions
Solve x² − 5x + 6 = 0.
- a = 1, b = −5, c = 6
- Discriminant: (−5)² − 4(1)(6) = 25 − 24 = 1 (positive, two solutions)
- x = (5 ± √1) / 2 = (5 ± 1) / 2
- x = 6/2 = 3 or x = 4/2 = 2
Example 2: One Repeated Solution
Solve x² − 6x + 9 = 0.
- a = 1, b = −6, c = 9
- Discriminant: 36 − 36 = 0 (one solution)
- x = 6 / 2 = 3
- The parabola touches the x-axis at its vertex.
Example 3: No Real Solutions
Solve x² + 1 = 0.
- a = 1, b = 0, c = 1
- Discriminant: 0 − 4 = −4 (negative, no real solutions)
- x = ±√(−4) / 2 = ±2i / 2 = ±i
- The solutions are complex: x = i and x = −i
Solution: a = 2, b = 7, c = −4. Discriminant = 49 + 32 = 81. x = (−7 ± 9) / 4. x₁ = 2/4 = 0.5, x₂ = −16/4 = −4.
When to Use the Quadratic Formula
Three main methods exist for solving quadratics: factoring, completing the square, and the quadratic formula. Here's how to choose:
- Factoring: Fast and elegant when the coefficients are small integers that factor nicely. Example: x² + 5x + 6 = (x + 2)(x + 3) = 0 gives x = −2 or −3 immediately.
- Completing the Square: Useful when you need the vertex form of a parabola. Best for deriving the quadratic formula itself or graphing.
- Quadratic Formula: Universal — works for any quadratic. Best when factoring is difficult, when coefficients involve decimals, or when you need exact irrational answers.
On tests, start by checking if factoring works quickly. If not, switch to the formula.
Real-World Applications
Quadratic equations model many physical phenomena:
Projectile Motion: The height h of an object thrown upward is h = −16t² + vt + s (where t is time, v is initial velocity, and s is starting height). To find when the object hits the ground, set h = 0 and solve for t.
Area Problems: Finding dimensions when the area is known often leads to quadratics. Example: A rectangular garden with area 24 ft² and length 2 feet more than the width gives x(x + 2) = 24, which is x² + 2x − 24 = 0.
Profit Optimization: Revenue often follows a parabolic pattern, and finding the maximum or minimum requires finding the vertex of a quadratic.
A ball is thrown upward from a height of 6 feet with an initial velocity of 40 ft/sec. Its height h after t seconds is h = −16t² + 40t + 6. When does the ball hit the ground?
Solution: Set h = 0: −16t² + 40t + 6 = 0. Using the quadratic formula: t = (−40 ± √(1600 + 384)) / −32 = (−40 ± √1984) / −32. The positive solution is approximately 2.64 seconds.
Common Mistakes to Avoid
- Sign errors with b: Remember that if b is negative, −b is positive. If b = −5, then −b = 5.
- Forgetting the ±: The formula always gives two solutions (unless the discriminant is zero). Don't stop after finding the first one.
- Dividing incorrectly: The entire numerator is divided by 2a, not just part of it. Use parentheses: (−b ± √D) / (2a).
- Misidentifying coefficients: Make sure the equation is in standard form ax² + bx + c = 0 before reading off a, b, and c.
Summary and Key Takeaways
The quadratic formula is your universal tool for solving ax² + bx + c = 0. Memorize it: x = (−b ± √(b² − 4ac)) / 2a. The discriminant (b² − 4ac) tells you whether you have two real solutions, one repeated solution, or no real solutions.
Practice with a variety of equations until the formula becomes automatic. With this single formula, you can solve any quadratic — something no other single method can claim.
For automated calculations and verification, use our Quadratic Equation Solver to check your work.