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For any equation \(ax^2 + bx + c = 0\), the solutions are \(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\). The part under the square root, \(b^2 – 4ac\), is the discriminant. It tells you whether there are two, one, or no real solutions.
How to Use the Quadratic Formula Calculator
- Type your equation, like
2x^2 + 3x - 5 = 0. You can leave off “= 0” or put terms on both sides, like3x^2 = 12. - Press Solve.
- See the exact answer (with fractions and square roots), the decimals, the vertex, and a graph of the parabola.
The Discriminant
| If \(b^2 – 4ac\) is… | Number of real solutions | The graph… |
|---|---|---|
| Positive | Two | crosses the x-axis twice |
| Zero | One (repeated) | touches the x-axis once |
| Negative | None (two complex solutions) | never touches the x-axis |
Worked Examples
Example 1: \(x^2 – 5x + 6 = 0\)
Here \(a = 1\), \(b = -5\), \(c = 6\). The discriminant is \(25 – 24 = 1\). So \(x = \frac{5 \pm 1}{2}\), which gives \(x = 2\) or \(x = 3\).
Example 2: \(2x^2 + 3x – 5 = 0\)
\(b^2 – 4ac = 9 + 40 = 49\). So \(x = \frac{-3 \pm 7}{4}\), which gives \(x = 1\) or \(x = -\frac{5}{2}\).
Example 3: \(x^2 – 2x – 1 = 0\)
\(b^2 – 4ac = 4 + 4 = 8\), and \(\sqrt{8} = 2\sqrt{2}\). So \(x = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2}\). As decimals, \(x \approx 2.414\) and \(x \approx -0.414\).
Example 4: \(x^2 + 2x + 5 = 0\)
\(b^2 – 4ac = 4 – 20 = -16\). The square root of a negative number uses \(i = \sqrt{-1}\): \(\sqrt{-16} = 4i\). So \(x = \frac{-2 \pm 4i}{2} = -1 \pm 2i\).
Where the Formula Comes From
The quadratic formula comes from completing the square on \(ax^2 + bx + c = 0\). Divide by \(a\), move \(c\) to the right, add \(\left(\frac{b}{2a}\right)^2\) to both sides, and take the square root. The result is the formula above.
The Vertex
The parabola \(y = ax^2 + bx + c\) has its turning point at \(x = -\frac{b}{2a}\). It opens up when \(a > 0\) and down when \(a < 0\). The solutions are always the same distance from the vertex on each side.
Common Mistakes
- Forgetting the negative in \(-b\). If \(b = -5\), then \(-b = 5\).
- Squaring a negative \(b\) wrong. \((-5)^2 = 25\), not \(-25\).
- Dividing only part of the top by \(2a\). The whole numerator, \(-b \pm \sqrt{b^2-4ac}\), is divided by \(2a\).
- Not moving everything to one side first. For \(3x^2 = 12\), write \(3x^2 – 12 = 0\) before finding \(a\), \(b\), and \(c\).
Learn the Math Behind the Quadratic Formula Calculator
The quadratic formula solves any equation ax² + bx + c = 0, including the characteristic equations of matrices and differential equations.
Frequently Asked Questions
When should I use the quadratic formula instead of factoring?
Use it any time. It always works. Factoring is faster only when the numbers factor nicely. If you are not sure, the formula is the safe choice. Try the factoring calculator to compare.
What does it mean if the discriminant is negative?
There are no real solutions. The parabola does not cross the x-axis. The two solutions are complex numbers that include \(i\).
Can a quadratic have just one solution?
Yes. When the discriminant is 0, both solutions are the same number. The graph touches the x-axis at its vertex.
Related Guides
- Factoring calculator
- Derivative calculator to find the slope of a parabola
- How to study calculus on your own