quadratic equations §5.5. objectives by the end of today, you should be able to… solve quadratic...
TRANSCRIPT
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QUADRATIC EQUATIONS§5.5
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OBJECTIVES
By the end of today, you should be able to…
• Solve quadratic equations by factoring and graphing.
What does it mean to solve an equation?
Quick Review:When you solve an equation, you’re finding the values that make the statement true.
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INVESTIGATION
Solve the following equations:
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QUADRATIC EQUATIONS
The standard form of a quadratic equation is ax2 + bx + c = 0, where a ≠ 0.
You can solve some quadratic equations in standard form by factoring the quadratic expression and then using…
The Zero-Product Property!
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THE ZERO-PRODUCT PROPERTY
The Zero-Product Property helps us solve quadratic equations that have been factored.
The Zero-Product Property states:
If ab = 0, then a = 0 or b = 0.
For example, if (x + 3)(x - 7)=0, then (x + 3) = 0 or (x - 7) = 0.
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EXAMPLE 1: SOLVING BY FACTORING
Solve by factoring.Step 1) Write in standard form.
Step 2) Factor.
Step 3) Use the Zero-Product Property.
Step 4) Solve for x.
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EXAMPLE 2: SOLVING BY FACTORING
Solve by factoring.Step 1) Write in standard form.
Step 2) Factor.
Step 3) Use the Zero-Product Property.
Step 4) Solve for x.
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EXAMPLE 3: SOLVING BY FACTORING
Solve by factoring.Step 1) Write in standard form.
Step 2) Factor.
Step 3) Use the Zero-Product Property.
Step 4) Solve for x.
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EXAMPLE 4: SOLVING BY FACTORING
Solve by factoring.Step 1) Write in standard form.
Step 2) Factor.
???
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SOLVING BY GRAPHING
You can solve a quadratic equation in standard form by graphing its related quadratic function,
y = ax2 + bx +c.
Quadratic equation cannot
be solved by factoring
Quadratic equation can be solved by graphing!
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SOLVING BY GRAPHING
Equation Solutions Vertex X-intercept Y-intercept
x2 – x – 6
x2 + 8x + 7
x2 – 6x + 9
x2 - 16
x2 + 4x – 5
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SOLVING BY GRAPHING
If we want to know what values of x make ax2 + bx +c = 0, where should we look?
We want to find where y = 0!
• When the graph of the function intersects the x-axis, the value of the function is zero, and each x-value is a zero of the function.
• A zero of a function is a solution of the equation ax2 + bx +c = 0.
A.K.A: Find the
x-intercepts!
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EXAMPLE 4: SOLVING BY GRAPHING
Solve by graphing.Step 1) Graph the related function, y = x2 + 6x + 4.
Step 2) Use the CALC feature to find the two zeros of the function.
Round to the nearest
thousandth!
x = -5.236 x = -0.764
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EXAMPLE 5: SOLVING BY GRAPHING
Solve by graphing.Step 1) Graph the related function, y = x2 – 5x + 2.
Step 2) Use the CALC feature to find the two zeros of the function.
Round to the nearest
thousandth!
x = 0.438 x = 4.562
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EXAMPLE 6: SOLVING BY GRAPHING
Solve by graphing.Step 1) Graph the related function, y = -x2 – 2x + 7 or y = x2 + 2x – 7.
Step 2) Use the CALC feature to find the two zeros of the function.
Round to the nearest
thousandth!
x = 1.828x = -3.828
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QUADRATIC EQUATIONSPT. 2
§5.5
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OBJECTIVES
By the end of today, you should be able to…
• Simplify square roots.
• Solve quadratic equations by finding square roots.
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SOLVE.What do you remember?...
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Solve by factoring: (Pick 4)
x2 – 6x – 7 = 0x2 + 6 = 7x9x2 – 8x = 1
x2 + 5x + 4x2 + 6 = 5x2x2 – x = 6
Solve:
X2 – 12 = 025x2 – 9 = 0
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SOLVING QUADRATIC EQUATIONS
If the quadratic equation is missing a middle term, or a b term, we can solve the equation in a
different way.
We can solve the equation by taking the square root of
each side.
Why?
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EXAMPLE 1: SIMPLIFYING RADICALS
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EXAMPLE 2: SIMPLIFYING RADICALS
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EXAMPLE 3: SIMPLIFYING RADICALS
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EXAMPLE 4: SOLVING BY FINDING SQUARE ROOTS
Solve by factoring.
5x 2 1800
Step 1) Rewrite in form ax2 = c.
Step 2) Isolate x2.
Step 3) Simplify.
Step 4) Take the square root of each side.
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EXAMPLE 5: SOLVING BY FINDING SQUARE ROOTS
Solve by factoring.
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EXAMPLE 6: SOLVING BY FINDING SQUARE ROOTS
Solve by factoring.
4x 2 250
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EXAMPLE 6: SOLVING BY FINDING SQUARE ROOTS
Solve by factoring.
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EXAMPLE 7: REAL WORLD CONNECTION-FIREFIGHTINGSmoke jumpers are in free fall from the time they jump out of a plane until they open their parachutes. The function y = -16t2 + 1600 models a jumpers height y in feet at t seconds for a jump from 1600 ft. How long is a jumper in free fall if the parachute opens at 1000 ft?
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Step 1) Substitute 1000 for y.
Step 2) Isolate t2.
Step 3) Take the square root of each side.
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EXAMPLE 8: FALLING OBJECTSThe function y = -16x2 + 270 models the height y in feet of Wile E. Coyote x seconds after he runs off the edge of a cliff that is 270 feet tall. How long does it take Wile E. Coyote to hit the ground?
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Step 1) Substitute 0 for y.
Step 2) Isolate x2.
Step 3) Take the square root of each side.
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CHALLENGENow it’s time for you to be the teacher! Write a quadratic equation with the given solutions.
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HOMEWORK
• p.266 (7-19)