Download 9.5 Using the Quadratic Formula

Algebra
9.5 Solving Quadratic Equations
Using the Quadratic Formula
This is an important section as there are many questions on the
STAR test about the quadratic formula.
Solve
2x2 + 10 = 28
- 10 -10
2x2 = 18
x2 = 9
x=+ 3
These are the solutions/roots of the equation.
We did not need the quadratic formula to solve this quadratic
equation because it was in the form…
Ax2 + C = # where b = 0.
What is the quadratic formula?
It is a formula used to solve any
quadratic equation in the form…
ax2 + bx + c = 0 when a ≠ 0 and
b2 – 4ac ≥ 0.
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Using the formula will produce the
solutions(roots) of the equation.
Here it is…try to memorize it…

x=
-b +
b2 – 4ac
2a
Quadratic+Formula. mp3
http://www.mathmadness.org/resources/Quadratic+Formula.mp3
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This formula can be used to find the
roots of any quadratic equation in the
form ax2 + bx + c = 0.
Find the roots of 2x2 + 10 = 28 using the quadratic
Remember the roots were x = + 3
formula…
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The equation must be in the form ax2 + bx + c = 0 before using the
quadratic formula.
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Must be 0 in order to use the quadratic formula.
2x2 + 10 = 28
+
-28 -28
144
X=
2x2 - 18 = 0 a = 2, b = 0, c = -18
4


-b +
b2 – 4ac
X=
2a
-0 +
+ 12
X=
4
(0)2 – 4(2)(-18)
X=
2(2)
X = 3 and - 3
These are the roots of the equation.
Find the roots of -3x2 + 4x = -5 using the quadratic
formula…
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
-3x2 + 4x = -5
+5 +5
-3x2 + 4x + 5 = 0
-4 +
16 + 60
X=
2(-3)
76
-4 +
-b +
b2 – 4ac
X=
a = -3, b = 4, c = 5
X=
(4)2 – 4(-3)(5)
X=
2(-3)
-4 + 2
19
X=
Both numbers in the numerator
must have common factors
of the denominator.
These are X =
the two
roots.
19
4
-6
2a
-4 +
76
Can you reduce?
Yes, by -2.
-6
2 +
2 -
19
19
and
3
3
You try! Find the roots of 4x2 - x = 7 using the
quadratic formula… who can do it on the board?
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

4x2 - x = 7
-7 -7
4x2 - x - 7 = 0
1 +
1 + 112
X=
8
1 +
-b +
b2 – 4ac
X=
1 +
a = 4, b = -1, c = -7
Can you reduce?
(-1)2 – 4(4)(-7)
No.
X=
2(4)
113
and
X=
2a
1 -
113
8
8
These are
the two
roots of the equation.
Note: This is on the STAR test.
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The roots of a quadratic equation are
the x-intercepts of the graph of the
quadratic (parabola).
The roots = x-intercepts
You try! Find the x-intercepts of the graph of y = x2 + 5x – 6.
using the quadratic formula… who can do it on the board?
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
y = x2 + 5x – 6
0 = x2 + 5x – 6
-b +
b2 – 4ac
X=
(5)2 – 4(1)(-6)
X=
2(1)
(substitute 0 for y)
a = 1, b = 5, c = -6
2a
-5 +
-5 +
49
X=
2
-5 + 7
-5 - 7
and
X=
2
X=
2
2
and
2
X= 1
-12
2
and
-6
These are
the two roots and
x-intercepts.
One from the HW
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P. 536 #46
HW
P. 536-537 # 33-36, 42-46, 53-55
Leave answers in simplified radical form.
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