4.10 Discriminant and parabolas

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.

 

Practice questions

Question 1

Question 2

Question 3

Question 4