Quadratic Formula Calculator

The quadratic formula solves equations of the form \(ax^2 + bx + c = 0\). The solution is given by the formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Example Calculation

Given the quadratic equation \(2x^2 + 5x - 3 = 0\), we can solve for \(x\) using the quadratic formula. Here:

• \(a = 2\)
• \(b = 5\)
• \(c = -3\)

The discriminant is calculated as:
\[ b^2 - 4ac = 5^2 - 4 \times 2 \times (-3) = 25 + 24 = 49 \]

This yields two solutions:
\[ x_1 = \frac{-5 + \sqrt{49}}{2 \times 2} = \frac{-5 + 7}{4} = \frac{2}{4} = 0.5 \]
\[ x_2 = \frac{-5 - \sqrt{49}}{2 \times 2} = \frac{-5 - 7}{4} = \frac{-12}{4} = -3 \]

Thus, the roots of the quadratic equation are \(x_1 = 0.5\) and \(x_2 = -3\).

Common FAQs

1. What does the discriminant tell us about the roots?

   • The discriminant (\(b^2 - 4ac\)) indicates the nature of the roots. If it's positive, there are two distinct real roots. If zero, one real root. If negative, two complex conjugate roots.

2. Can the quadratic formula always solve any quadratic equation?

   • Yes, the quadratic formula provides a solution for any quadratic equation, including those with complex roots.

3. How do you handle quadratic equations with fractional or irrational coefficients?

   • The quadratic formula remains applicable regardless of coefficient types, as long as the values are real or complex numbers.