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.
Using the formula will produce the solutions(roots) of the equation.
Here it is…try to memorize it…
x =
This formula can be used to find the roots of any quadratic equation in the form ax2 + bx + c = 0.
b2 – 4ac
2a
-b +
http://www.mathmadness.org/resources/Quadratic+Formula.mp3
Quadratic+Formula.mp3
Find the roots of 2x2 + 10 = 28 using the quadratic formula…
The equation must be in the form ax2 + bx + c = 0 before using the quadratic formula.
2x2 + 10 = 28 -28 -28 2x2 - 18 = 0
Remember the roots were x = + 3
Must be 0 in order to use the quadratic formula.
-b + b2 – 4ac
2aX =
-0 + (0)2 – 4(2)(-18)
2(2)X =
+ 144
4X =
+ 12
4X =
3 and - 3X =
a = 2, b = 0, c = -18
These are the roots of the equation.
Find the roots of -3x2 + 4x = -5 using the quadratic formula…
-3x2 + 4x = -5 +5 +5 -3x2 + 4x + 5 = 0
-b + b2 – 4ac
2aX =
-4 + (4)2 – 4(-3)(5)
2(-3)X =
a = -3, b = 4, c = 5
-4 + 16 + 60
2(-3)X =
-4 + 76
-6X =
76
4 19
-4 + 2 19
-6X =
Can you reduce?
Both numbers in the numerator must have common factorsof the denominator.
2 + 19
3X =These are
the two roots.
and 2 - 19
3
Yes, by -2.
You try! Find the roots of 4x2 - x = 7 using the quadratic formula… who can do it on the board?
4x2 - x = 7 -7 -7 4x2 - x - 7 = 0
-b + b2 – 4ac
2aX =
1 + (-1)2 – 4(4)(-7)
2(4)X =
a = 4, b = -1, c = -7
1 + 1 + 112
8X =
1 + 113
8X =
These arethe two roots of the equation.
1 - 113
8
and
Can you reduce?
No.
Note: This is on the STAR test.
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?
y = x2 + 5x – 6 0 = x2 + 5x – 6 (substitute 0 for y)
-b + b2 – 4ac
2aX =
-5 + (5)2 – 4(1)(-6)
2(1)X =
a = 1, b = 5, c = -6
-5 + 49
2X =
-5 + 7
2X =
These arethe two roots and x-intercepts.
-5 - 7
2
and
2
2X =
-12
2
and
1X = -6 and
One from the HW
P. 536 #46
HW
P. 536-537 # 33-36, 42-46, 53-55
Leave answers in simplified radical form.