Quadratic Equation Calculator

Solve any quadratic equation with step-by-step solutions and graphical visualization

Enter the coefficients for \(ax^2 + bx + c = 0\)

Equation: \(x^2 + 0x + 0 = 0\)

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x:

\(ax^2 + bx + c = 0\)

where a, b, and c are constants, and a ≠ 0.

The Quadratic Formula

The solutions to a quadratic equation are given by:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

where the discriminant \(b^2 - 4ac\) determines the nature of the roots.

Understanding the Discriminant

The discriminant \(b^2 - 4ac\) tells us about the nature of the roots:

  • If \(b^2 - 4ac > 0\), there are two distinct real roots
  • If \(b^2 - 4ac = 0\), there is one repeated real root
  • If \(b^2 - 4ac < 0\), there are two complex conjugate roots

Examples

Basic Example

Solve: \(x^2 - 5x + 6 = 0\)

Using the quadratic formula:

\(x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2}\)

Therefore, \(x = 3\) or \(x = 2\)

One Root Example

Solve: \(x^2 - 6x + 9 = 0\)

Using the quadratic formula:

\(x = \frac{6 \pm \sqrt{36 - 36}}{2} = \frac{6 \pm 0}{2} = 3\)

Therefore, \(x = 3\) (repeated root)

Complex Roots Example

Solve: \(x^2 + x + 1 = 0\)

Using the quadratic formula:

\(x = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2}\)

Therefore, \(x = -\frac{1}{2} \pm \frac{\sqrt{3}}{2}i\)

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which gives the solutions to the quadratic equation \(ax^2 + bx + c = 0\).

How do I know if a quadratic equation has real solutions?

A quadratic equation has real solutions if its discriminant \(b^2 - 4ac\) is greater than or equal to zero. If the discriminant is negative, the equation has complex solutions.

What does it mean when a quadratic equation has one solution?

When a quadratic equation has exactly one solution, it means the discriminant \(b^2 - 4ac = 0\). This results in a repeated root, and the parabola touches the x-axis at exactly one point.

Can I solve a quadratic equation by factoring?

Yes, if the quadratic expression can be factored, you can solve the equation by setting each factor equal to zero. However, not all quadratic expressions can be easily factored, which is why the quadratic formula is a more general solution method.

Real-Life Applications of Quadratic Equations

Projectile Motion

When an object is thrown or launched, its path is a parabola. Quadratic equations help us predict this path, calculating things like maximum height and distance traveled. For example, think of a basketball player shooting for the hoop – the ball's trajectory is a perfect example of a quadratic equation in action.

Optimization

Businesses and individuals often need to find the best way to do something – maximize profit, minimize cost, or use resources most efficiently. Quadratic equations can model these situations, helping find the "sweet spot." For instance, a company might use a quadratic equation to determine the optimal price for a product to maximize revenue.

Engineering

From the graceful curve of a bridge's suspension cables to the shape of a satellite dish, parabolas (and thus, quadratic equations) are fundamental in engineering. These shapes have unique properties. For example, a parabolic reflector focuses all incoming light or sound waves to a single point, which is why satellite dishes are that shape.

Area Calculation

Imagine you have a fixed length of fencing and want to enclose the largest possible rectangular area. This classic problem can be solved using a quadratic equation. The relationship between the sides of the rectangle and the enclosed area leads to a quadratic equation, and finding its maximum point reveals the dimensions that maximize the area.