Quadratic Equations

Report
QUADRATIC
EQUATIONS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
Basics
• 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
factoring and using The Principle of Zero
Products
If ab = 0, then either a = 0, b = 0, or both a
and b = 0.
Ex: Solve (4t + 1)(3t – 5) = 0
Notice the equation as given is of the form ab = 0
 set each factor equal to 0 and solve
4t + 1 = 0 Subtract 1
Divide by 4
4t = – 1
t=–¼
3t – 5 = 0
3t = 5
t = 5/3
Add 5
Divide by 3
Solution: t = - ¼ and 5/3  t = {- ¼, 5/3}
Ex: Solve x2 + 7x + 6 = 0
Quadratic equation  factor the left hand side (LHS)
x2 + 7x + 6 = (x + 6 )(x + 1 )
 x2 + 7x + 6 = (x + 6)(x + 1) = 0
Now the equation as given is of the form ab = 0
 set each factor equal to 0 and solve
x+6=0
x=–6
x+1=0
x=–1
Solution: x = - 6 and – 1  x = {-6, -1}
Ex: Solve x2 + 10x = – 25
Quadratic equation but not of the form ax2 + bx + c = 0
Add 25  x2 + 10x + 25 = 0
Quadratic equation  factor the left hand side (LHS)
x2 + 10x + 25 = (x + 5 )(x + 5 )
 x2 + 10x + 25 = (x + 5)(x + 5) = 0
Now the equation as given is of the form ab = 0
 set each factor equal to 0 and solve
x+5=0
x+5=0
x=–5
x=–5
Solution: x = - 5  x = {- 5}  repeated root
Ex: Solve 12y2 – 5y = 2
Quadratic equation but not of the form ax2 + bx + c = 0
Subtract 2  12y2 – 5y – 2 = 0
Quadratic equation  factor the left hand side (LHS)
ac method  a = 12 and c = – 2
ac = (12)(-2) = - 24  factors of – 24 that sum to - 5
1&-24, 2&-12, 3&-8, . . . 
 12y2 – 5y – 2 = 12y2 + 3y – 8y – 2
= 3y(4y + 1) – 2(4y + 1)
= (3y – 2)(4y + 1)
 12y2 – 5y – 2 = 0
 12y2 – 5y – 2 = (3y - 2)(4y + 1) = 0
Now the equation as given is of the form ab = 0
 set each factor equal to 0 and solve
3y – 2 = 0
4y + 1 = 0
3y = 2
4y = – 1
y = 2/3
y=–¼
Solution: y = 2/3 and – ¼  y = {2/3, - ¼ }
Ex: Solve 5x2 = 6x
Quadratic equation but not of the form ax2 + bx + c = 0
Subtract 6x  5x2 – 6x = 0
Quadratic equation  factor the left hand side (LHS)
5x2 – 6x = x( 5x – 6 )
 5x2 – 6x = x(5x – 6) = 0
Now the equation as given is of the form ab = 0
 set each factor equal to 0 and solve
5x – 6 = 0
x=0
5x = 6
x = 6/5
Solution: x = 0 and 6/5  x = {0, 6/5}
Solving by taking square roots
• An alternate method of solving a quadratic
equation is using the Principle of Taking
the Square Root of Each Side of an
Equation
If x2 = a, then x = + a
Ex: Solve by taking square roots 3x2 – 36 = 0
First, isolate x2: 3x2 – 36 = 0
3x2 = 36
x2 = 12
Now take the square root of both sides:
x  12
x   12
2
x   223
x2 3
Ex: Solve by taking square roots 4(z – 3)2 = 100
First, isolate the squared factor:
4(z – 3)2 = 100
(z – 3)2 = 25
Now take the square root of both sides:
(z  3)  25
2
z  3   25
z–3=+5
z=3+5
 z = 3 + 5 = 8 and z = 3 – 5 = – 2
Ex: Solve by taking square roots 5(x + 5)2 – 75 = 0
First, isolate the squared factor:
5(x + 5)2 = 75
(x + 5)2 = 15
Now take the square root of both sides:
( x  5 )2  15
x  5   15
x  5  15
x  5  15 , x  5  15
Completing the Square
• Recall from factoring that a Perfect-Square
Trinomial is the square of a binomial:
Perfect square Trinomial
x2 + 8x + 16
x2 – 6x + 9
Binomial Square
(x + 4)2
(x – 3)2
• The square of half of the coefficient of x
equals the constant term:
( ½ * 8 )2 = 16
[½ (-6)]2 = 9
Completing the Square
• Write the equation in the form x2 + bx = c
• Add to each side of the equation [½(b)]2
• Factor the perfect-square trinomial
x2 + bx + [½(b)] 2 = c + [½(b)]2
• Take the square root of both sides of the
equation
• Solve for x
Ex: Solve w2 + 6w + 4 = 0 by completing the square
First, rewrite the equation with the constant on one
side of the equals and a lead coefficient of 1.
w2 + 66w = – 4
Add [½(b)]2 to both sides: b = 6  [½(6)]2 = 32 = 9
w2 + 6w + 9 = – 4 + 9
w2 + 6w + 9 = 5
(w + 3)2 = 5
Now take the square root of both sides
( w  3)  5
2
w 3  5
w  3  5
w  {3  5,3  5}
Ex: Solve 2r2 = 3 – 5r by completing the square
First, rewrite the equation with the constant on one
side of the equals and a lead coefficient of 1.
2r2 + 5r = 3
 r2 + (5/2)r
(5/2) = (3/2)
Add [½(b)]2 to both sides: b = 5/2 [½(5/2)]2 = (5/4)2
= 25/16
r2 + (5/2)r + 25/16 = (3/2) + 25/16
r2 + (5/2)r + 25/16 = 24/16 + 25/16
(r + 5/4)2 = 49/16
Now take the square root of both sides
(r  5 / 4)  49 /16
2
r  5 / 4  (7 / 4)
r  (5 / 4)  (7 / 4)
r = - (5/4) + (7/4) = 2/4 = ½
and r = - (5/4) - (7/4) = -12/4 = - 3
r = { ½ , - 3}
Ex: Solve 3p – 5 = (p – 1)(p – 2)
Is this a quadratic equation? FOIL the RHS
3p – 5 = p2 – 2p – p + 2
3p – 5 = p2 – 3p + 2
p2 – 6p + 7 = 0
Collect all terms
A-ha . . .
Quadratic Equation  complete the square
p2 – 6p = – 7
p2 – 6p + 9 = – 7 + 9
(p – 3)2 = 2
 [½(-6)]2 = (-3)2 = 9
(p  3)  2
2
p3  2
p  3 2
p  {3  2,3  2}
The Quadratic Formula
• Consider a quadratic equation of the form
ax2 + bx + c = 0 for a nonzero
• Completing the square
ax  bx   c
b
c
x  x
a
a
b
b
c
b
x  x
 
a
4a
a 4a
2
2
2
2
2
2
2
The Quadratic Formula
b
b
4ac
b
x  x


a
4a
4a
4a
b  b  4ac

x  
2a 
4a

2
2
2
2
2
2
2
2
2
Solutions to ax2 + bx + c = 0 for a nonzero
are
2
 b  b  4ac
x
2a
Ex: Use the Quadratic Formula to solve1x2 + 7x
7 +6=0
Recall: For quadratic equation ax2 + bx + c = 0,
the solutions to a quadratic equation are given by
 b  b 2  4ac
x
2a
Identify a, b, and c in ax2 + bx + c = 0:
a=1
b= 7
c= 6
Now evaluate the quadratic formula at the identified
values of a, b, and c
 7  7 2  4(1)(6)
x
2(1)
 7  49  24
x
2
 7  25
x
2
75
x
2
x = ( - 7 + 5)/2 = - 1 and x = (-7 – 5)/2 = - 6
x = { - 1, - 6 }
Ex: Use the Quadratic Formula to solve
2
2m2 + 1m – 10 = 0
Recall: For quadratic equation ax2 + bx + c = 0,
the solutions to a quadratic equation are given by
 b  b 2  4ac
m
2a
Identify a, b, and c in am2 + bm + c = 0:
a=2
b= 1
c = - 10
Now evaluate the quadratic formula at the identified
values of a, b, and c
 1  12  4(2)(10)
m
2(2)
 1  1  80
m
4
 1  81
m
4
1 9
m
4
m = ( - 1 + 9)/4 = 2 and m = (-1 – 9)/4 = - 5/2
m = { 2, - 5/2 }
Any questions . . .

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