3.2 Quadratic Equations
A quadratic equation in one variable is any equation equivalent to the standard form:
ax2+bx+c=0 a≠0, b, c are real numbers
Graph of a parabola!
Zero­Product Property: If AB=0 then A=0 or B=0.
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To solve an equation using the zero­product property:
1) Put the equation into standard form. Factor completely.
2) Set each factor to zero and solve. Collect all solutions.
Ex. x2+5x=6
x2+5x­6=0
x+6=0 or x­1=0
(x+6)(x­1)=0
x = ­6,1
Zero­Product Property: If AB=0 then A=0 or B=0.
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1) 1 + 2x(x­9) = 10­x
2) 3) (3x­1)(x+4) = x+9
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Suppose the weekly demand for a certain product is given by the equation p=90­2x, where x is the number of units sold and p is the price per unit in dollars. How many units must be sold in order for the weekly revenue to be $1000? Note: There can be more than one answer.
Revenue = Demand * Price
R=
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Square root property:
(x+1)2­2=7
Complete the square:
If x2=a then x=±
(x+1)2=9
x2+bx = k x+1=±3
(divide both sides by coefficient of x2 if needed)
+(b/2)
+(b/2)2
x2+bx = k 2
Factor
x = ­4,2
Add (b/2)2 to both sides.
(x+b/2)2 = k +(b/2)2
Now it can be solved using the square root property!
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(CTS) Complete the square: x2+bx = k (divide both sides by coefficient of x2 if needed)
+(b/2)2
+(b/2)2
x2+bx = k Factor
Add (b/2)2 to both sides.
(x+b/2)2 = k +(b/2)2
Solve by CTS and the square root property!
1) x2+6x­7 = ­12
2) 2x2­28x = 13
3) 3x2­9x = 5
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Deriving the quadratic formula:
ax2+bx+c = 0
(a≠0)
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Solve with the quadratic formula:
ax2+bx+c = 0
(a≠0)
x=
1) x2+4x = ­11
2) x(3x­7) + 21 = 14
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2
Within the quadratic formula, Δ=b -4ac is known as the ________________ .
1) If Δ=0 then the quadratic equation has one real solution (called a double root).
2) If Δ>0 then the quadratic equation has two real solutions.
2) If Δ<0 then the quadratic equation has two (nonreal) complex (conjugate) solutions.
When Δ is a perfect square (like 9 or 1/4 for example) then the quadratic equation has two rational solutions and could be solved more efficiently by factoring!
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You try:
1) Solve for x: (x­1)(x­2)=12
2) Determine the discriminant for the following equation and how many solutions of what type there are: y2+7y+5=4. Then solve it.
3) Jack throws a rock off of a cliff with an initial downward velocity of 2 meters per second. The height in meters of the rock after t seconds is modeled roughly by h(t)=­5t2­2t+125. How long does it take to hit the bottom of the cliff? Hint: Set h(t)=0.
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