6.02 Discriminant

Quadratic equations and parabolas

We have seen when solving quadratic equations that there can be two, one or no real solutions. Let's think about the graphs of quadratics:

No solutions	One solution	Two solutions

 

As shown above there are only three possibilities, with respect to $x$x intercepts, when graphing quadratics. These possibilities are as follows:

• No real solutions. This means there are no $x$x-intercepts or real zeros.  
• One real solution. This is where the two zeros are actually equal, and the parabola has one $x$x-intercept where it just touches the $x$x-axis at the turning point.
• Two real solutions. These are the two distinct zeros or $x$x-intercepts, where the quadratic passes through the $x$x-axis.

The solutions to a quadratic equation correspond to the $x$x values that occur when $y=0$y=0 in a quadratic function, and these are the places where a function crosses the $x$x-axis.

 

The discriminant

We have revised a range of algebraic techniques to solve quadratic equations, and obviously, if we are able to find these actual solutions, we can answer the question of how many solutions or roots a quadratic has. But there is a faster way!

Let's look again at the quadratic formula:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$x=−b±√b2−4ac2a​

Specifically, let's look at what happens if the square root part $\sqrt{b^2-4ac}$√b2−4ac takes on different values...

$b^2-4ac<0$b2−4ac<0  or  $b^2-4ac=0$b2−4ac=0  or  $b^2-4ac>0$b2−4ac>0 

• If $b^2-4ac<0$b2−4ac<0, then the square root is negative, and we know that we cannot take the square root of a negative number and get real solutions. This is the case where we have zero real roots.
• If $b^2-4ac=0$b2−4ac=0, then the square root is $0$0 and then the quadratic equation becomes just $x=\frac{-b}{2a}$x=−b2a​. and we have a single root. This is actually the equation for the axis of symmetry.
• If $b^2-4ac>0$b2−4ac>0, then the square root will have two values, one for $+\sqrt{b^2-4ac}$+√b2−4ac and one for $-\sqrt{b^2-4ac}$−√b2−4ac. The quadratic formula will then generate for us two distinct real roots. If this value is a square number, we will have two rational roots, seeing as the square root of a square number will lead us with a rational result. For any other positive number, we will end up with two roots with surds in them. 

This expression $b^2-4ac$b2−4ac within the quadratic formula is called the discriminant, and it determines the number of real solutions a quadratic function will have. The symbol $\Delta$Δ is used as a shortcut for$b^2-4ac$b2−4ac .

 

 

Remember that every quadratic function can be sketched, even if it has no real roots. Every quadratic function is a parabola and has a vertex, which means that it has either a maximum or minimum value which can be found. Knowing the value of the discriminant, and the value of $a$a in $y=ax^2+bx+c$y=ax2+bx+c which determines its concavity, can give us enough information for a rough sketch of the parabola.

Use the following applet to explore how the discriminant relates to the number of solutions to a quadratic equation.

Created with Geogebra

 

Practice questions

Question 1

Question 2

Question 3

Question 4