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```					Quadratic Equations,

• by factorization
• by graphical method
• by taking square roots
By taking square roots

x k
2

x  k
2

x k
Remember standard form for a quadratic equation is:

ax  bx  c  0 ax  c  0
2
0x                          2

In this form we could have the case where b = 0.

When this is the case, we get the x2 alone and then square
root both sides.

2x  6  0
2               Get x2 alone by adding 6 to both sides and then
dividing both sides by 2
+6    +6
Now take the square root of both
2x  6
2
x  3
2
sides remembering that you must
positive and
consider both the positive and
2         2                       negative root.
negative root.

x 3
Let's
check:        2
2 3 6  0                     2
2  3 6  0
66  0                    66  0
By taking square roots
(2 x  3)  4
2

2x  3  4
?
2x  3  2
2x  5
x  2.5
A quadratic equation must contain two roots.
By taking square roots
(2 x  3)  4
2

2x  3   4
2x  3  2
2 x  5 or 1
x  2.5 or 0.5
By taking square roots
3x  1  8
2
x 3
2

3x  8  1
2

x  3
2

3x  9
2

x 3
2
3x   9

3   3
By taking square roots
5  2 x  11
2
 2x   2
6

2    2
 2 x  11  5
2

x  3
2

 2x  6
2

No solution, x² cannot be negative
Exercise 9F Page 298
ax  bx  c  0
2
0
What if in standard form, c = 0? We could factor by pulling
an x out of each term.

2 x  3x  0
2
Factor out the common x

x2 x  3  0     Use the Null Factor law and set each
factor = 0 and solve.

x  0 or 2x  3  0
3
x  0 or x          If you put either of these values in for x
in the original equation you can see it
2       makes a true statement.
By factorization
x  7 x  10  0
2

( x  5)(x  2)  0
x  5  0 _ or _ x  2  0
x  5 _ or _ x  2
roots (solutions)
A quadratic equation is an equation equivalent to one of the form
ax  bx  c  0
2

Where a, b, and c are real numbers and a  0
So if we have an equation in x and the highest power is 2, it is quadratic.
To solve a quadratic equation we get it in the form above
and see if it will factor.
x  5x  6
2                     Get form above by subtracting 5x and
adding 6 to both sides to get 0 on right side.
-5x + 6     -5x + 6

x 2  5x  6  0         Factor.

x  3x  2  0           Use the Null Factor law and set each
factor = 0 and solve.

x  3  0 or x  2  0                  x 3          x2
ax  bx  c  0
2

What are we going to do if we have non-zero values for
a, b and c but can't factor the left hand side?

x  6x  3  0
2                     This will not factor so we will complete the
square and apply the square root method.

x  6 x  3
2                    First get the constant term on the other side by
subtracting 3 from both sides.

x  6 x  ___  3  ___
2
9          9                        x2  6x  9  6

Let's add 9. Right now we'll see that it works and then we'll look at how
to find it.
x2  6x  9  6                   Now factor the left hand side.

x  3x  3  6             This can be written as:    x  3  2
6
Now we'll get rid of the square by
two identical factors        square rooting both sides.

x  3  2
 6
Remember you need both the
positive and negative root!

x3  6                  Subtract 3 from both sides to get x alone.

These are the answers in exact form. We
x  3  6                can put them in a calculator to get two