Quadratic Equation Calculator

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form \(ax^2 + bx + c = 0\), where \(a \neq 0\). The word "quadratic" comes from the Latin word "quadratus," which means "square." These equations appear everywhere in science and engineering, from projectile motion in physics to profit optimization in business.

The Quadratic Formula

The most reliable method for solving any quadratic equation is the quadratic formula:

The expression under the square root, \(b^2 - 4ac\), is called the discriminant (D). It tells you the nature of the roots:

• If \(D > 0\): Two distinct real roots
• If \(D = 0\): One repeated real root (the parabola touches the x-axis at one point)
• If \(D < 0\): Two complex conjugate roots (the parabola does not cross the x-axis)

Step-by-Step Example

Solve \(3x^2 - 12x + 9 = 0\):

1. Identify: \(a = 3\), \(b = -12\), \(c = 9\)
2. Discriminant: \(D = (-12)^2 - 4(3)(9) = 144 - 108 = 36\)
3. Since \(D > 0\), there are two real roots
4. \(x_1 = \frac{12 + 6}{6} = 3\) and \(x_2 = \frac{12 - 6}{6} = 1\)

How to use this calculator

1. Enter the coefficients \(a\), \(b\), and \(c\) from your equation.
2. Make sure the equation is written in standard form, \(ax^2 + bx + c = 0\).
3. Click solve to see the roots, discriminant, and step-by-step work.
4. Use the graph to check where the parabola crosses the x-axis, if it does.

The formula explained

The quadratic formula computes the solutions, or roots, of any quadratic equation in standard form. The discriminant \(b^2 - 4ac\) tells you how many real solutions the equation has.

• a = the coefficient of \(x^2\)
• b = the coefficient of \(x\)
• c = the constant term
• x = the unknown value that makes the equation true
• \(b^2 - 4ac\) = the discriminant, which determines the type of solutions

Step by step method

1. Write the equation in standard form \(ax^2 + bx + c = 0\).
2. Identify \(a\), \(b\), and \(c\), then compute the discriminant \(b^2 - 4ac\).
3. Substitute the values into \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) and simplify.

Worked example

Problem. Solve \(2x^2 - 3x - 2 = 0\).

1. Here, \(a=2\), \(b=-3\), and \(c=-2\). The discriminant is \((-3)^2 - 4(2)(-2) = 9 + 16 = 25\).
2. Use the quadratic formula: \(x = \frac{-(-3) \pm \sqrt{25}}{2(2)} = \frac{3 \pm 5}{4}\).
3. So the two solutions are \(x = \frac{8}{4} = 2\) and \(x = \frac{-2}{4} = -\frac{1}{2}\).

Tips and common mistakes

• Always move every term to one side first so the equation matches \(ax^2 + bx + c = 0\).
• If the discriminant is negative, the equation has no real roots, so the graph does not cross the x-axis.

Frequently Asked Questions