Quadratic Formula

Before we look at the quadratic formula, let’s take a quick look at the reason behind its importance and fame — quadratic equations.

Quadratic Equations — A Quick Intro

A quadratic equation is a second–degree equation with only one variable.

The standard form of a quadratic equation is —

ax^2 + bx + c = 0, \hspace{0.75em} a \neq 0

Quadratic Equation — Standard Form

The leading coefficient $a \hspace{0.2em}$ cannot be $\hspace{0.2em} 0 \hspace{0.2em}$ because then the second–degree term ( $\hspace{0.2em} x^2$ –term) would vanish and our equation would no longer be a second–degree equation.

So $\hspace{0.2em} x^2 + 2x - 3 = 0 \hspace{0.2em}$ is an example of a quadratic equation.

But a quadratic equation does not always have to be in the standard form. As long as an equation can be rearranged and written as $\hspace{0.2em} ax^2 + bx + c = 0, \hspace{0.5em} a \neq 0 \hspace{0.2em}$ , it is a quadratic equation.

Here are a few examples of quadratic equations.

Equation	In Standard Form
$\hspace{0.2em} -4x + 3x^2 - 1 = 0 \hspace{0.2em}$	$\hspace{0.2em} {\color{Red} 3} x^2 {\color{Teal} \hspace{0.2em} - \hspace{0.2em} 4} x {\color{Orchid} \hspace{0.2em} - \hspace{0.2em} 1} = 0 \hspace{0.2em}$
$\hspace{0.2em} 7x - 5x^2 = 0 \hspace{0.2em}$	$\hspace{0.2em} {\color{Red} -5} x^2 {\color{Teal} \hspace{0.2em} + \hspace{0.2em} 7} x {\color{Orchid} \hspace{0.2em} + \hspace{0.2em} 0} = 0 \hspace{0.2em}$
$\hspace{0.2em} 2x^2 - 16 = 0 \hspace{0.2em}$	$\hspace{0.2em} {\color{Red} 2} x^2 {\color{Teal} \hspace{0.2em} + \hspace{0.2em} 0} x {\color{Orchid} \hspace{0.2em} - \hspace{0.2em} 16} = 0 \hspace{0.2em}$

Note — If you don't see a coefficient in a term, it's $\hspace{0.2em} 1 \hspace{0.2em}$ or $\hspace{0.2em} -1 \hspace{0.2em}$ depending on the sign of the term. For example, in the equation $\hspace{0.2em} x^2 - x + 5 = 0 \hspace{0.2em}$ , $\hspace{0.2em} a = 1 \hspace{0.2em}$ and $\hspace{0.2em} b = -1 \hspace{0.2em}$ .

Solving Quadratic Equations

Solving a quadratic equation means finding the values of $\hspace{0.2em} x \hspace{0.2em}$ (or whatever the variable is) for which the equation becomes true.

Values of $\hspace{0.2em} x \hspace{0.2em}$ that satisfy the equation are the solutions of the equation.

For example, the solutions of the quadratic equation $\hspace{0.2em} x^2 - 3x + 2 = 0 \hspace{0.2em}$ are $\hspace{0.2em} 1 \hspace{0.2em}$ and $\hspace{0.2em} 2 \hspace{0.2em}$ .

Now, there are three popular methods of solving quadratic equations – factoring, completing the square, and using the quadratic formula.

As is obvious from the title, in this tutorial we’d focus on how to solve quadratic equations using the quadratic formula.

The Quadratic Formula

The quadratic formula tells us, the solutions of a quadratic equation, $\hspace{0.2em} ax^2 + bx + c = 0 \hspace{0.2em}$ , are given by —

x_1, \hspace{0.4em} x_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Solutions of a quadratic equation

In the formula above, we have a $\hspace{0.2em} \pm \hspace{0.2em}$ sign between $\hspace{0.2em} -b \hspace{0.2em}$ and the radical. So, what does it mean?

Well, quadratic equations typically have two solutions. Using the plus sign gives us one solution and using the minus sign gives us the other.

So, if $\hspace{0.2em} x_1 \hspace{0.2em}$ and $\hspace{0.2em} x_2 \hspace{0.2em}$ are the two solutions of a quadratic equation —

x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

The following example should make things clearer.

Example

Using the quadratic formula, solve $\hspace{0.2em} x^2 + 2x - 3 = 0 \hspace{0.2em}$ .

Solution

The Discriminant

In the quadratic formula, the part under the radical is known as the discriminant. It is usually denoted by the symbol, $\hspace{0.2em} \Delta \hspace{0.2em}$ (delta). So,

\Delta \hspace{0.2em} = \hspace{0.2em} b^2 - 4ac

At this point, you might be wondering why we care about the discriminant at all.

What Makes the Discriminant Important?

The value of the discriminant informs us of the nature of the quadratic equations roots (or solutions).

$\hspace{0.2em} \Delta > 0 \hspace{0.2em}$	Two distinct real solutions
$\hspace{0.2em} \Delta = 0 \hspace{0.2em}$	Repeated real solution
$\hspace{0.2em} \Delta < 0 \hspace{0.2em}$	Two distinct non-real solutions

Let me explain using the following examples.

Example

Without actually solving the quadratic equation $\hspace{0.2em} x^2 - 5x + 7 = 0 \hspace{0.2em}$ , find the nature of its roots. Then solve the equation.

Solution

Example

Calculate the discriminant for the quadratic equation $\hspace{0.2em} x^2 - 5x + 7 = 0 \hspace{0.2em}$ and hence, find the nature of its roots. Thereafter, solve the equation.

Solution

Example

Using its discriminant, find the nature of roots of the quadratic equation $\hspace{0.2em} x^2 - 5x + 7 = 0 \hspace{0.2em}$ . Also, solve the equation.

Solution

TK – when to use the quadratic formula….??

And with that, we come to the end of this tutorial on the quadratic formula. Until next time.