What Is b^2-4ac? Quadratic Formula Explained

[askor]

As we know that $x_{1,2}$ of a quadratic function can be found with the below formula:

$\frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

What do you call the $b^2 - 4ac$?

 

[462chevelle]

It's the discriminant, if I spelled that right.
Its value let's you know if you have complex or real solutions.

 

[Mark44]

It's the discriminant, if I spelled that right.

Yes, you spelled it correctly.

Its value let's you know if you have complex or real solutions.

It discriminates between two real solutions, one real and repeated solution, and two complex solutions, depending on whether the discriminant is positive, zero, or negative, respectively.

 

[askor]

I wasn't able to find about discriminant in my Calculus textbook.

What book I can found about this discriminant?

 

[bhillyard]

I would look in an algebra book

 

[Mondayman]

I wasn't able to find about discriminant in my Calculus textbook.

What book I can found about this discriminant?

You should be able to find this in any algebra book. Look under quadratic equations.

 

[symbolipoint]

You should be able to find this in any algebra book. Look under quadratic equations.

Yes, that is right. Any intermediate or college algebra textbook will discuss the discriminant of a quadratic equation or of a quadratic expression.

The discriminant occurs when you use Completing the Square to generally solve a quadratic equation; as well as if you use Completing the Square to solve a particular quadratic equation.

 

[Mark44]

The discriminant occurs when you use Completing the Square to generally solve a quadratic equation; as well as if you use Completing the Square to solve a particular quadratic equation.

The discriminant shows up in the Quadratic Formula, which is derived by completing the square. If you solve a quadratic equation by completing the square, you won't see the discriminant.

For example, solve $x^2 - 4x - 1 = 0$
1. By Quadratic Formula
$\Rightarrow x = \frac{4 \pm \sqrt{4^2 - (4\cdot 1 \cdot (-1)}}{2} = \frac{4 \pm \sqrt{20}}{2} = 2 \pm \sqrt{5}$
Here the discriminant is $b^2 - 4ac$ = 16 - (-4) = 20

2. By completing the square
$x^2 - 4x - 1 = 0$
$\Rightarrow x^2 - 4x + 4 = 1 + 4$
$\Rightarrow (x - 2)^2 = 5$
$\Rightarrow x - 2 = \pm \sqrt{5}$
$\Rightarrow x = 2 \pm \sqrt{5}$

Notice that the discriminant (20) never explicitly appears in completing the square.

 

[symbolipoint]

Mark44 shows the ordinary algebra step behavior, that we usually simplify from one step to the next, and we do not then see the uncomputed expression for the discriminant. If we WANTED to, we could leave that part uncomputed, and finish its computation last.