SOLVING QUADRATICS EQUATIONS

With Professor OwlCreated by Robbie Smith

Solve by Graphing

Quadratic Term: ax²Linear Term: bxConstant Term: c

In order to have a solution, the line or

parabola must touch the x-axis once or twice. If it

doesn’t touch at all, there is no solution.

You must also find the vertex and axis

of symmetry

Solve by Graphing

To find the points, you can make a table!

X Y

-4 6

-3 5

-2 6

-1 9

0 14

Ex. x²+6x+14=y

Then graph the results.

This graph shows that there is no solution.

Vertex: (-3,5)Axis or Sym: -3Solution: None

Solve by Factoring

Remember: Most equations can be factored, but not all equations can be factored.

Here are some examples of factoring.

Ex. (x+5)(3x-2)=0X+5=0 3x-2=0 -5 -5 +2 +2X=-5 3x=2

3 3x=2 3

Unlike the last example which was already factored, you must factor

this one.Ex. 16x² -9=0

(4x+3)(4x-3)=04x+3=0 4x-3=0

-3 -3 +3 +34x=-3 4x=34 4 4 4X=-3 x=3 4 4

Solve by Factoring More Examples

Set the equation equal to zero

Ex. x²+16=8x -8x -8x x²-8x+16=0 (x-4)(x-4)=0

x-4=0 +4 +4

x=4

Write in Proper FormEx. 8x² -2=-2x +2x +2x

8x² -2=0x²-16=0

(8x+4)(8x-4) 8

(x+4)(8x-4)x=-4 x=1 2

Ex. 3x²+6x=0 3x(x+2)=0

x= 0 x= -2

Completing the Square

Examples of Perfect Squares

x²+4x+4 (x+2)(x+2)=(x+2)²x²-8+16 (x-4)(x-4)=(x-4)²x²+14x+49 (x+7)(x+7)=(x+7)²

To find the constant, do the following:Divide Linear by 2

Then square it.x²+4x+?

4/2=22²=4

x²+4x+4

Completing the Square

Examples

x²+4-10=0 +10 +10

x²+4x=104/2=2 2²=4

x²+4x+4=10+4(x+2)²=14

√(x+2)²=√14x+2=√14

x=-2+√14 x=-2-√14

3x²+6x-9=03 3 3 3x²+2x-3=0

x²+2x+1=3+1√(x+1)=√4

x+1=2x+1=-2

x=1x=-3

Solve by the Quadratic Formula When you get an equation, it looks like this:

ax²+bx+cWhen using the quadratic formula, use this

formula:x=-b±√b²-4ac

2a

Let’s see an example!

The Discriminant: b²-4ac (Very Important)

It tells you the numbers, root, and solutions. Sweet!

Solve by the Quadratic Formula

Discriminant:Negative-2 Imaginary SolutionsZero- 1 Real SolutionPositive-perfect Square- 2 Reals RationalPositive-Non-perfect square- 2 Reals Irrational

You must find the discriminant!

3x²+6x-9=06²-4(3)(-9)36+108=144144: Two Reals Rational

x=-6±122(3)

x=-6+12 6

x=-6-12 6

x=1 x=-3

And That’s It!