Quadratic Formula Calculator: Solving Quadratic Equations

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree, typically written in the form:

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

Where:

• \(a\), \(b\), and \(c\) are constants (numbers)
• \(a \neq 0\) (if \(a = 0\), the equation would be linear, not quadratic)
• \(x\) is the variable we're solving for

The Quadratic Formula

The quadratic formula is a powerful tool that allows us to solve any quadratic equation. It is expressed as:

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

Where:

• \(x\) represents the solutions (roots) of the equation
• \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\)
• The \(\pm\) symbol indicates that there are typically two solutions

Calculation Steps

1. Identify the values of \(a\), \(b\), and \(c\) from your quadratic equation.
2. Calculate the discriminant: \(b^2 - 4ac\)
3. Substitute the values into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
4. Simplify the expression under the square root.
5. Calculate the two values of \(x\) using the \(+\) and \(-\) versions of the \(\pm\) symbol.
6. Simplify your answers if possible.

Example with Visual Representation

Let's solve the quadratic equation: \(2x^2 - 7x + 3 = 0\)

Here, \(a = 2\), \(b = -7\), and \(c = 3\)

Applying the quadratic formula:

\[x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(2)(3)}}{2(2)}\]

\[x = \frac{7 \pm \sqrt{49 - 24}}{4} = \frac{7 \pm \sqrt{25}}{4} = \frac{7 \pm 5}{4}\]

Therefore, the solutions are:

\[x_1 = \frac{7 + 5}{4} = \frac{12}{4} = 3\]

\[x_2 = \frac{7 - 5}{4} = \frac{2}{4} = \frac{1}{2}\]

The graph above shows the parabola \(y = 2x^2 - 7x + 3\). The x-intercepts (where the curve crosses the x-axis) are the roots of the equation: \(x = \frac{1}{2}\) and \(x = 3\).