Derivation of the Sum of Roots

 

Derivation of the Product of Roots

By difference of two squares:

The formulas

sum of roots: −b/a
product of roots: c/a 

As you can see from the derivation below, when you are trying to solve aquadratic equations in the form of ax²+bx +c. The sum and product of the roots can be rewritten using the two formulas above.

Example 1

The example below illustrates how this formula applies to the quadratic equation x² + 5x +6. As you can see the sum of the roots is indeed -b/a and the product of the roots is c/a.

Example 2

The example below illustrates how this formula applies to the quadratic equation x² - 2x - 8. Again, both formulas--for the sum and the product boil down to -b/a and c/a, respectively.

What is the discriminant anyway?

Answer : The discriminant is a number that can be calculated from any quadratic equation

A quadratic equation is an equation that can be written as

ax ² + bx + c where a ≠ 0

What is the formula for the Discriminant?

The discriminant in a quadratic equation is found by the following formula and the discriminant provides critical information regarding the nature of the roots/solutions of any quadratic equation.

discriminant= b² − 4ac

Example of the discriminant

• Quadratic equation = y = 3x² + 9x + 5
• The discriminant = 9 ² − 4 • 3 •5

Imortant pre requisites

To understand what the discriminant does, it's important that you have a good understanding of

What a quadratic equation is:

Answer: A quadratic equation is an equation in the form of y=ax2+bx+c where ane0 Read more here on this topic

What does the graph of a quadratic equation look like:

Answer: A parabola.

What is the solution of a quadratic equation:

Answer: The solution can be thought of in two different ways.

Algebraically, the solution occurs when y = 0 . So the solution is wherey=ax2+bx+c becomes 9=ax2+bx+c

Graphically, since y = 0 is the x-axis, the solution is where the parabola intercepts the x-axis. (This only works for real solutions).
In the picture below, the left parabola has 2 real solutions (red dots), the middle parabola has 1 real solution (red dot) and the right most parabola has no real solutions (yes, it does have imaginary ones)

What does this formula tell us?

Answer: The discriminant tells us the following information about a quadratic equation:

• If the solution is a real number or an imaginary number.
• If the solution is rational or if it is irrational.
• If the solution is 1 unique number or two different numbers

Nature of the Solutions

Value of the discriminant	Type and number of Solutions	Example of graph
Positive Discriminant b² − 4ac > 0	Two Real Solutions If the discriminant is a perfect square the roots are rational. Otherwise, they areirrational.
Positive and a Perfect Square b² − 4ac = perfect square	Two Real RationalSolutions.
Positive and anot a perfect square b² − 4ac = not a perfect square	Two Real IrrationalSolutions.
Discriminant is Zero b² − 4ac = 0	One Real Solution
Negative Discriminant b² − 4ac < 0	No Real Solutions Two ImaginarySolutions

• Example 1
  Quadratic Equation: y = x² + 2x + 1
  • a = 1
  • b = 2
  • c = 1
  The discriminant for this equation is

  b2−4ac22−4(1)(1)=0 

  Since the discriminant is zero, there should be 1 real solution to this equation. Below is a picture representing the graph and one solution of this quadratic equation

  Graph of y = x² + 2x + 1

  

Zeros of Quadratic Function

How to find the zeros of functions; tutorial with examples and detailed solutions. The zeros of a function f are found by solving the equation f(x) = 0.

Example 1: Find the zero of the linear function f is given by

f(x) = -2 x + 4

Solution to Example 1

To find the zeros of function f, solve the equation

f(x) = -2x + 4 = 0

Hence the zero of f is give by

x = 2

Example 2: Find the zeros of the quadratic function f is given by

f(x) = -2 x ² - 5 x + 7

Solution to Example 2

Solve f(x) = 0

f(x) = -2 x ² - 5 x + 7 = 0

Factor the expression -2 x ² - 6 x + 8

(-2x - 7)(x - 1) = 0

and solve for x

x = -7 / 2 and x = 1

The graph of function f is shown below. The zeros of a function are the x coordinates of the x intercepts of the graph of f.

Graphing Quadratic Equations

A Quadratic Equation in Standard Form
(a, b, and c can have any value, except that a can't be 0.) 

Here is an example:

Graphing

You can graph a Quadratic Equation using our Function Grapher, but to really understand what is going on, you can make the graph yourself. Read On!

The Simplest Quadratic

The simplest Quadratic Equation is:

f(x) = x²

And its graph is simple too:

This is the curve f(x) = x²
It is a parabola.

Now let us see what happens when we introduce the "a" value:

f(x) = ax²

• Larger values of a squash the curve
• Smaller values of a expand it
• And negative values of a flip it upside down

Play With It Now is a good time to play with the "Quadratic Equation Explorer" so you can see what different values of a, b and c produce

The "General" Quadratic

Before graphing we rearrange the equation, from this:

f(x) = ax² + bx + c

To this:

f(x) = a(x-h)² + k

Where:

h = -b/2a
k = f( h )

In other words, calculate h (=-b/2a), then find k by calculating the whole equation for x=h

First of all ... Why?

Well, the wonderful thing about this new form is that h and k show you the very lowest (or very highest) point, called the vertex: And also the curve is symmetrical (mirror image) about the axis that passes through x=h, making it easy to graph

So ...

• h shows you how far left (or right) the curve has been shifted from x=0
• k shows you how far up (or down) the curve has been shifted from y=0

Lets see an example of how to do this:

Example: Plot f(x) = 2x² - 12x + 16

First, let's note down:

• a = 2,
• b = -12, and
• c = 16

Now, what do we know?

• a is positive, so it is an "upwards" graph ("U" shaped)
• a is 2, so it is a little "squashed" compared to the x² graph

Next, let's calculate h:

h = -b/2a = -(-12)/(2x2) = 3

And next we can calculate k (using h=3):

k = f(3) = 2(3)² - 12·3 + 16 = 18-36+16 = -2

So now we can plot the graph (with real understanding!):

We also know: the vertex is (3,-2), and the axis is x=3

From A Graph to The Equation

What if you have a graph, and want to find an equation?

Example: you have just plotted some interesting data, and it looks Quadratic:

Just knowing those two points we can come up with an equation.

Firstly, we know h and k (at the vertex):

(h, k) = (1,1)

So let's put that into this form of the equation:

f(x) = a(x-h)² + k

f(x) = a(x-1)² + 1

Then we calculate "a":

We know (0, 1.5) so:		f(0) = 1.5
And we know the function (except for a):		f(0) = a(0-1)2 + 1 = 1.5
Simplify:		f(0) = a + 1 = 1.5
		a = 0.5

And so here is the resulting Quadratic Equation:

f(x) = 0.5(x-1)² + 1

Note: This may not be the correct equation for the data, but it’s a good model and the best we can come up with.

Quadratic Equations

An example of a Quadratic Equation:

The name Quadratic comes from "quad" meaning square, because the variable getssquared (like x²).

It is also called an "Equation of Degree 2" (because of the "2" on the x)

The Standard Form of a Quadratic Equation looks like this:

a, b and c are known values. a can't be 0. "x" is the variable or unknown (you don't know it yet).

Here are some more examples:

		In this one a=2, b=5 and c=3
		This one is a little more tricky: Where is a? In fact a=1, as we don't usually write "1x2" b = -3 And where is c? Well, c=0, so is not shown.
		Oops! This one is not a quadratic equation, because it is missing x2 (in other words a=0, and that means it can't be quadratic)

Hidden Quadratic Equations!

So the "Standard Form" of a Quadratic Equation is

ax² + bx + c = 0

But sometimes a quadratic equation doesn't look like that! For example:

In disguise	→	In Standard Form	a, b and c
x2 = 3x -1	Move all terms to left hand side	x2 - 3x + 1 = 0	a=1, b=-3, c=1
2(w2 - 2w) = 5	Expand (undo the brackets), and move 5 to left	2w2 - 4w - 5 = 0	a=2, b=-4, c=-5
z(z-1) = 3	Expand, and move 3 to left	z2 - z - 3 = 0	a=1, b=-1, c=-3
5 + 1/x - 1/x2 = 0	Multiply by x2	5x2 + x - 1 = 0	a=5, b=1, c=-1

Have a Play With It I have a "Quadratic Equation Explorer" so you can see: the graph it makes, and the solutions (called "roots").

How To Solve It?

The "solutions" to the Quadratic Equation are where it is equal to zero. There are usually 2 solutions (as shown in the graph above).

They are also called "roots", or sometimes "zeros"

There are 3 ways to find the solutions:

1. You can Factor the Quadratic (find what to multiply to make the Quadratic Equation)

2. You can Complete the Square, or

3. You can use the special Quadratic Formula:

Just plug in the values of a, b and c, and do the calculations.

We will look at this method in more detail now.

About the Quadratic Formula

Plus/Minus

First of all what is that plus/minus thing that looks like ± ?

The ± means there are TWO answers: Here you see why you can get two answers:

But sometimes you don't get two real answers, and the "Discriminant" shows why ...

Discriminant

Do you see b² - 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

when b² - 4ac is positive, you get two Real solutions
when it is zero you get just ONE real solution (both answers are the same)
when it is negative you get two Complex solutions

I will explain about Complex solutions after you have seen how to use the formula.

Using the Quadratic Formula

Just put the values of a, b and c into the Quadratic Formula, and do the calculations.

Example: Solve 5x² + 6x + 1 = 0

Coefficients are:		a = 5, b = 6, c = 1
Quadratic Formula:		x = [ -b ± √(b2-4ac) ] / 2a
Put in a, b and c:		x = [ -6 ± √(62-4×5×1) ] / (2×5)
Solve:		x = [ -6 ± √(36-20) ]/10
		x = [ -6 ± √(16) ]/10
		x = ( -6 ± 4 )/10
		x = -0.2 or -1

Answer: x = -0.2 or x = -1 And you can see them on this graph.

Check -0.2:	5×(-0.2)² + 6×(-0.2) + 1  = 5×(0.04) + 6×(-0.2) + 1  = 0.2 -1.2 + 1  = 0
Check -1:	5×(-1)² + 6×(-1) + 1  = 5×(1) + 6×(-1) + 1  = 5 - 6 + 1  = 0

Remembering The Formula

I don't know of an easy way to remember the Quadratic Formula, but a kind reader suggested singing it to "Pop Goes the Weasel":

♫	"x equals minus b		♫	"All around the mulberry bush
	plus or minus the square root			The monkey chased the weasel
	of b-squared minus four a c			The monkey thought 'twas all in fun
	all over two a"			Pop! goes the weasel"

Try singing it a few times and it will get stuck in your head!

Complex Solutions?

When the Discriminant (the value b² - 4ac) is negative you get Complex solutions ... what does that mean?

It means your answer will include Imaginary Numbers. Wow!

Example: Solve 5x² + 2x + 1 = 0

Coefficients are:		a = 5, b = 2, c = 1
Note that The Discriminant is negative:		b2 - 4ac = 22 - 4×5×1 = -16
Use the Quadratic Formula:		x = [ -2 ± √(-16) ] / 10
The square root of -16 is 4i (i is √-1, read Imaginary Numbers to find out more)
So:		x = ( -2 ± 4i )/10

Answer: x = -0.2 ± 0.4i The graph does not cross the x-axis. That is why we ended up with complex numbers.

In some ways it is actually easier ... you don't have to calculate the solutions, just leave it as -0.2 ± 0.4i.

Summary

Quadratic Equation in Standard Form: ax² + bx + c = 0
Quadratic Equations can be factored
Quadratic Formula: x = [ -b ± √(b²-4ac) ] / 2a
When the Discriminant (b²-4ac) is:
• positive, there are 2 real solutions
• zero, there is one real solution
• negative, there are 2 complex solutions