Quadratic Equation Calculator

What is a Quadratic Equation?

A quadratic equation is like a special math sentence that describes a curved line called a parabola. It's a bit like describing the path a ball takes when you throw it up in the air!

How to Solve a Quadratic Equation

Solving a quadratic equation is like finding where the ball hits the ground. We use a special formula called the quadratic formula to find these points, which we call roots or solutions.

Formula

The quadratic equation looks like this:

\[ ax^2 + bx + c = 0 \]

And we solve it using the quadratic formula:

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

Here's what each part means:

• \(a\), \(b\), and \(c\) are numbers in the quadratic equation
• \(x\) is what we're solving for (where the ball hits the ground)
• \(\pm\) means we'll get two answers: one with + and one with -
• \(\sqrt{}\) means square root (like asking, "what number times itself gives us this?")

The Discriminant

An important part of the quadratic formula is the expression under the square root: \(b^2 - 4ac\). This is called the discriminant.

The discriminant tells us about the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is one repeated real root.
• If the discriminant is negative, there are two complex roots.

Calculation Steps

1. Identify \(a\), \(b\), and \(c\) in your equation
2. Calculate \(b^2\) (b times itself)
3. Multiply 4, \(a\), and \(c\)
4. Subtract the result of step 3 from \(b^2\) to get the discriminant
5. Find the square root of the discriminant
6. Add this to -\(b\) for one solution, subtract for the other
7. Divide both results by 2\(a\)

Example and Visual Representation

Let's solve: \(x^2 - 5x + 6 = 0\)

Here, \(a=1\), \(b=-5\), and \(c=6\)

Using the quadratic formula, we get:

\[ x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2} \]

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

Let's show this with a picture:

In this picture, the blue curve is our quadratic equation. The red dots show where the curve crosses the x-axis. These points are at x = 2 and x = 3, which are our solutions!