Last updated:
Quadratic Formula
A quadratic equation is any second‑degree polynomial equation written in standard form:
ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
To solve for x, you can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
The symbol ± means there are two solutions–one using addition and the other using subtraction. The term under the square root, b² - 4ac, is called the discriminant and determines how many (and what type of) solutions exist.
The calculator above automates this process. Enter the coefficients of any quadratic and it instantly computes the discriminant and gives you the exact roots.
How Do You Solve a Quadratic Equation Using the Formula?
Applying the quadratic formula is straightforward if you follow these steps:
- Put the equation in standard form: make sure all terms are on one side and zero is on the other, and write them in descending powers of x. Example: 3x = -x² + 5 becomes x² + 3x - 5 = 0.
- Identify a, b, and c: a = coefficient of x², b = coefficient of x, c = constant term. Watch the signs–they belong to the coefficients.
- Compute the discriminant D = b² - 4ac. This tells you what kind of roots to expect.
- Plug a, b, and the square root of D into the formula x = (-b ± √D) / (2a). Be careful with parentheses around -b.
- Simplify each solution separately: add and subtract √D for the two roots. Reduce fractions if possible.
Keep in mind: if a = 0, the equation is not quadratic (it becomes linear), and the formula is invalid.
The Discriminant and Root Types
The discriminant D = b² - 4ac acts as a “decision maker”:
- D > 0 → two distinct real roots. The quadratic crosses the x‑axis at two points.
- D = 0 → one repeated real root (a double root). The graph touches the x‑axis at the vertex.
- D < 0 → two complex conjugate roots of the form p ± qi. The graph does not intersect the x‑axis.
For example, the equation x² - 4x + 13 = 0 gives D = (-4)² - 4·1·13 = 16 - 52 = -36. The square root of -36 is 6i, and the solutions are (4 ± 6i)/2 = 2 ± 3i.
Understanding the discriminant helps you decide whether factoring is possible over integers (D must be a perfect square) and whether you need to report real or complex answers.
Worked Examples
1. Two real roots
Solve 2x² - 7x + 3 = 0.
Here a = 2, b = -7, c = 3.
Discriminant: D = (-7)² - 4·2·3 = 49 - 24 = 25.
√25 = 5.
Roots: x = (7 ± 5) / (4) → x = (7+5)/4 = 12/4 = 3, and x = (7-5)/4 = 2/4 = ½.
Solutions: 3 and 0.5.
2. One repeated root
Solve x² - 6x + 9 = 0.
a = 1, b = -6, c = 9.
D = 36 - 36 = 0.
x = (6 ± 0) / 2 = 3.
The double root is 3.
3. Complex roots
Solve 2x² + 2x + 1 = 0.
a = 2, b = 2, c = 1.
D = 4 - 8 = -4.
√(-4) = 2i.
x = (-2 ± 2i) / 4 = -½ ± ½i.
Roots: -0.5 + 0.5i and -0.5 - 0.5i.
Derivation of the Quadratic Formula
The formula comes from completing the square on the general quadratic ax² + bx + c = 0:
- Divide by a (since a ≠ 0): x² + (b/a)x = -c/a.
- Add (b/(2a))² to both sides: x² + (b/a)x + b²/(4a²) = -c/a + b²/(4a²).
- Left side becomes (x + b/(2a))². Right side combine over denominator 4a²: (b² - 4ac)/(4a²).
- Take square root: x + b/(2a) = ±√(b² - 4ac) / (2a).
- Isolate x: x = -b/(2a) ± √(b² - 4ac) / (2a) = (-b ± √(b² - 4ac)) / (2a).
This derivation explains why the discriminant appears and why a is in the denominator–dividing by a early on imposes a ≠ 0.
Common Mistakes When Applying the Formula
- Forgetting to set the equation to zero: The formula requires the right side to be 0. Rearrange first.
- Dropping negative signs: (-b) means the opposite sign of b. If b = 4, -b = -4; if b = -3, -b = +3.
- Misapplying the denominator: The entire numerator is divided by 2a, not just a part. Use parentheses: (-b ± √D) / (2a).
- Assuming the discriminant is always positive: Always calculate b² - 4ac before taking the square root. If it’s negative, the roots are complex and cannot be plotted on the real number line.
Whether you are solving a practical physics problem or an algebra homework question, the quadratic formula provides a reliable, always‑applicable method. Combine it with the discriminant to quickly judge the expected answers, and verify your work by substituting the roots back into the original equation.
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a). It gives the solutions (roots) of any quadratic equation of the form ax² + bx + c = 0, provided a ≠ 0. The expression under the square root, b² - 4ac, is called the discriminant.
What does the discriminant tell you?
The discriminant D = b² - 4ac determines the nature of the roots. If D > 0, there are two distinct real roots. If D = 0, one repeated real root. If D < 0, two complex conjugate roots. It also indicates if the quadratic can be factored over integers.
How can I check my quadratic formula answer?
Plug your solutions back into the original equation ax² + bx + c. Both should satisfy the equation (within rounding error). Alternatively, check that the sum of roots equals -b/a and the product equals c/a.
Do I always need to use the quadratic formula?
No. If the quadratic easily factors (e.g., x² - 5x + 6 = 0), factoring is faster. The quadratic formula works universally, making it the safest method when factoring is not obvious or when a, b, c are decimals or large numbers.
What is the axis of symmetry in a quadratic function?
For a quadratic function y = ax² + bx + c, the axis of symmetry is the vertical line x = -b/(2a). This value is the x-coordinate of the vertex; it’s also the average of the two roots, showing that the roots are symmetric around this line.