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Algebra 2
Section 3.5
Quadratic Equations
Equations such as: 5x2 + 8x – 12 = 0 , 9x2 – 25 = 0, and 3x2 + 4x = 0
are called quadratic equations. Each equation contains a polynomial of
the second degree (exponent of 2).
The standard form of a quadratic equation is: ax2 + bx + c = 0, where
a,b, and c are numbers and (a  0).
Examples: 3x2 + 5x + 6 = 0 or x2 – 2x + 1 = 0
One way to solve a quadratic equation is to:
- write the equation in standard form {ax2 + bx + c = 0}
- factor it
- and set each factor equal to 0
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Solve each equation.
1. x2 – 13x + 40 = 0
(x – 8)(x – 5) = 0 factor into 2 binomials
x – 8 = 0 or x – 5 = 0 set each factor equal to 0
+8 +8
x=8
+5 +5
or
x=5
2. 0 = x2 + 15x + 50
x2 + 15x + 50 = 0 switch around
(x + 10)(x + 5) = 0 factor into 2 binomials
x + 10 = 0 or x + 5 = 0 set each factor equal to 0
-10 -10
x = -10
-5
or
-5
x = -5
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3. 6 = b2 – b
-6
-6 subtract 6 to get in standard form
b2 – b – 6 = 0 standard form
(b – 3)(b + 2) = 0 factor into 2 binomials
b – 3 = 0 or b + 2 = 0 set each factor equal to 0
-2 -2
+3 +3
b=3
or
b = -2
4. 2a2 – 10a = 0
2a(a – 5) = 0 factor 2a out
2a = 0 or a – 5 = 0 set each factor equal to 0
+5 +5
a=0
or
a=5
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5. 15a = a2
-15a -15a subtract 15a from both sides
a2 – 15a = 0
a(a – 15) = 0 factor a out
a = 0 or a – 15 = 0 set each factor equal to 0
+15 +15
or
a = 15
6. 0 = c2 – 36
c2 – 36 = 0 switch around
(c + 6)(c – 6) = 0 c2 – 36 is a difference of 2 squares:
c + 6 = 0 or c – 6 = 0 set each factor equal to 0
-6
+6 +6
-6
c = -6
a2 – b2 = (a + b)(a – b)
or
c=6
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7. x4 – 26x2 + 25 = 0
(x2 – 25)(x2 – 1) = 0 factor into 2 binomials
x2 – 25 = 0 or x2 – 1 = 0 set each factor equal to 0
both of these are a difference of 2 squares: a2 – b2 = (a + b)(a – b)
(x + 5)(x – 5) = 0 or (x + 1)(x – 1) = 0
x + 5 = 0 or x – 5 = 0
x + 1 = 0 or x – 1 = 0
-5 -5
+5 +5
-1 -1
+1 +1
x = -5 or x = 5
x = -1 or x = 1
x = -5, 5, -1, 1
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8. 900 + x4 = 109x2
-109x2
-109x2
x4 – 109x2 + 900 = 0
(x2 – 100)(x2 – 9) = 0
x2 – 100 = 0
or
x2 – 9 = 0
(x + 10)(x – 10) = 0
(x + 3)(x – 3) = 0
x + 10 = 0 or x – 10 = 0 x + 3 = 0 or x – 3 = 0
x = -10 or x = 10
x = -3 or x = 3
x = -10, 10, -3, 3
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Solve each equation.
1.) a2 – a – 42 = 0
2.) y2 = 12 – y
3.) 6y = 16 – y2
4.) 3x2 = -12x
5.) z2 – 64 = 0
6.) 16 = z2
Algebra 2
Section 3.5
Assignment
7.) a4 – 29a2 + 100 = 0
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