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QUADRATIC EQUATIONS
MSJC ~ San Jacinto CampusMath Center Workshop Series
Theresa Hert
Simplify Radicals
Radicals with index 2 are referred to as square roots.
Simplify Radicals
Break down the radicand, the number inside the radical, into prime factors.
Circle a pair of matching factors, take out THE factor.
Since no operation sign is visible, the “glue” holding everything together is Multiplication.
When you bring a factor out of the radical, it gets multiplied to the number in front of the radical.
Simplify the Radical
5 63
5 3 3 7 5 3 7
15 7
Simplify Rational Expressions containing Radicals
First simplify the radical.
To reduce the fraction, Factor.
Beware of addition.
Plus sign – use one set of parentheses to factor out what is common.
Simplify this Rational Expression containing a Radical
9 45
6
9 3 3 5
6
3 3 1 5
6
3 5
2
9 3 5
6
Quadratic Equations
contain both an equal sign and
a variable with exponent 2.
General form: ax2 + bx + c = 0
• A quadratic equation is an equation equivalent to an equation of the type
ax2 + bx + c = 0, where a is nonzero
• We can solve a quadratic equation by using the Quadratic Formula
a2ac4bb
x2
The Quadratic Formula
• Solve the equation ax2 + bx + c = 0 for x by Completing the Square
2ax bx c
2b c
x xa a
2 2
2
2 2
b b c bx x
a 4a a 4a
The Quadratic Formula
Solutions to ax2 + bx + c = 0 for a nonzero are
22
2
b b 4acx2a 4a
2b b 4acx
2a
2 2
2
2 2 2
b b 4ac bx x
a 4a 4a 4a
Solve this Quadratic Equationby using the Quadratic Formula
6y2 – 3y – 5 = 0
a = 6 b = -3 c = -5
2( 3) ( 3) 4(6)( 5)
2(6)x
3 9 120
12x
6y2 – 3y – 5 = 0
a = 6 b = -3 c = -5
because of the addition,
you can NOT reduce the fraction
2( 3) ( 3) 4(6)( 5)
2(6)x
3 129
12x
3 9 120
12x
Ex: Use the Quadratic Formula to solve x2 + 7x + 6 = 0
Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by
a2ac4bb
x2
Identify a, b, and c in ax2 + bx + c = 0:
a = b = c = 1
1
7
7
6
6
Now evaluate the quadratic formula at the identified values of a, b, and c
)1(2)6)(1(477
x2
224497
x
2257
x
257
x
x = ( - 7 + 5)/2 = - 1 and x = (-7 – 5)/2 = - 6
x = { - 1, - 6 }
x2 + 7x + 6 = 0
a = 1 b = 7 c = 6
Ex: Use the Quadratic Formula to solve
2m2 + m – 10 = 0
Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by
a
acbbm
2
42
Identify a, b, and c in am2 + bm + c = 0:
a = b = c = 2
2
1
1
- 10
– 10
Now evaluate the quadratic formula at the identified values of a, b, and c
)2(2)10)(2(411
m2
48011
m
4811
m
491
m
m = ( - 1 + 9)/4 = 2 and m = (-1 – 9)/4 = - 5/2
m = { 2, - 5/2 }
2x2 + 1x – 10 = 0
a = 2 b = 1 c = -10
Ex: Use the Quadratic Formula to solve
x2 + 5x = -3
x2 + 5x + 3 = 0
2b b 4acx
2a
Identify a, b, and c in ax2 + bx + c = 0:
a = b = c = 1
3 1
+ 5
+ 5
Now evaluate the quadratic formula at the identified values of a, b, and c
3
25 5 4( 1 )( 3 )x
2( 1 )
5 25 12x
2
5 13x
2
5 13 5 13x and x
2 2
x2 + 5x + 3 = 0 a = 1 b = 5 c = 3
Ex: Use the Quadratic Formula to solve
10x2 – 5x = 0 10x2 – 5x + 0 = 0
2b b 4acx
2a
Identify a, b, and c in ax2 + bx + c = 0:
a = b = c = 10
0 10
- 5
- 5
Now evaluate the quadratic formula at the identified values of a, b, and c
0
2
5 5 4 10 0x
2 10
5 25 0x
20
5 5x
20
1x and x 0
2
10x2 – 5x + 0 = 0 a = 10 b = -5 c = 0
Solve: use the Quadratic Formula.
2
2
2x 6 x 1 0
a 2 b 6 c 1
6 6 4 2 1
2 2
6 36 8
4
6 28
4
6 2 2 7
4
6 2 7
4
2 3 7
4
3 7
2
