Transcript quadratic equation

Solving
Equations
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
-5x + 6
Get form above by subtracting 5x and
adding 6 to both sides to get 0 on right side.
-5x + 6
x 2  5x  6  0
x  3x  2  0
Factor.
Use the Null Factor law and set each
factor = 0 and solve.
x  3  0 or x  2  0
x3
x2
Remember standard form for a quadratic equation is:
2
2
ax  bx
0x  c  0
ax  c  0
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
Get x2 alone by adding 6 to both sides and then
dividing both sides by 2
2
+6
+6
2x  6
2
2
2
x 3
Now take the square root of both
sides remembering that you must
consider both the positive
positiveand
and
negative root.
negative
root.
x  3
2
Let's
check:
 
2
2 3 6  0
66  0


2
2  3 6  0
66  0
ax  bx  0
c0
2
What if in standard form, c = 0? We could factor by pulling
an x out of each term.
2 x  3x  0
Factor out the common x
x2 x  3  0
Use the Null Factor law and set each
factor = 0 and solve.
2
x  0 or 2x  3  0
3
x  0 or x 
2
If you put either of these values in for x
in the original equation you can see it
makes a true statement.
SUMMARY OF SOLVING QUADRATIC EQUATIONS
• Get the equation in standard form:
ax  bx  c  0
2
• If there is no middle term (b = 0) then get the x2 alone and square
root both sides (if you get a negative under the square root there are
no real solutions).
• If there is no constant term (c = 0) then factor out the common x
and use the null factor law to solve (set each factor = 0).
• If a, b and c are non-zero, see if you can factor and use the null
factor law to solve.
• If it doesn't factor or is hard to factor, use the quadratic formula
to solve (if you get a negative under the square root there are no real
solutions).
This "discriminates" or tells us what type of solutions we'll have.
ax  bx  c  0
2
 b  b  4ac
x
2a
2
If we have a quadratic equation and are considering solutions
from the real number system, using the quadratic formula, one of
three things can happen.
1. The "stuff" under the square root can be positive and we'd get
two unequal real solutions if b 2  4ac  0
2. The "stuff" under the square root can be zero and we'd get one
solution (called a repeated or double root because it would factor
2
if
b
 4us
acthe
 0same solution).
into two equal factors, each giving
3. The "stuff" under the square root can be negative and we'd get
no real solutions.
if b 2  4ac  0
The "stuff" under the square root is called the discriminant.
The Discriminant
  b 2  4ac