5-6 The Quadratic Formula
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Transcript of 5-6 The Quadratic Formula
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5-6 The Quadratic Formula
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You have learned several methods for solving quadratic equations: graphing, making tables, factoring, using square roots, and completing the square. Another method is to use the Quadratic Formula, which allows you to solve a quadratic equation in standard form.
By completing the square on the standard form of a quadratic equation, you can determine the Quadratic Formula.
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ax2 + bx + c = 0
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The symmetry of a quadratic
function is evident in the last
step, . These two
zeros are the same distance,
, away from the axis of
symmetry, ,with one zero
on either side of the vertex.
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You can use the Quadratic Formula to solve any quadratic equation that is written in standard form, including equations with real solutions or complex solutions.
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Find the zeros of f(x) = x2 + 3x – 7 using the Quadratic Formula.
Set f(x) = 0.
Write the Quadratic Formula.
Substitute 1 for a, 3 for b, and –7 for c.
Simplify.
Write in simplest form.
x2 + 3x – 7 = 0
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Find the zeros of f(x)= x2 – 8x + 10 using the Quadratic Formula.
x2 – 8x + 10 = 0
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Find the zeros of f(x) = 4x2 + 3x + 2 using the Quadratic Formula.
Quadratic Functions with Complex Zeros
Set f(x) = 0.
Write the Quadratic Formula.
Substitute 4 for a, 3 for b, and 2 for c.
Simplify.
Write in terms of i.
f(x)= 4x2 + 3x + 2
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Find the zeros of g(x) = 3x2 – x + 8 using the Quadratic Formula.
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The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.
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Make sure the equation is in standard form before you evaluate the discriminant, b2 – 4ac.
Caution!
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Find the type and number of solutions for the equation.
Analyzing Quadratic Equations by Using the Discriminant
x2 + 36 = 12x
x2 – 12x + 36 = 0
b2 – 4ac
(–12)2 – 4(1)(36)
144 – 144 = 0
b2 – 4ac = 0
The equation has one distinct real solution.
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Find the type and number of solutions for the equation.
Analyzing Quadratic Equations by Using the Discriminant
x2 + 40 = 12x
x2 – 12x + 40 = 0
b2 – 4ac
(–12)2 – 4(1)(40)
144 – 160 = –16
b2 –4ac < 0
The equation has two distinct nonreal complex solutions.
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Find the type and number of solutions for the equation.
Analyzing Quadratic Equations by Using the Discriminant
x2 + 30 = 12x
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Find the type and number of solutions for the equation.
x2 – 4x = –4
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Find the type and number of solutions for the equation.
x2 – 4x = –8
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The graph shows related functions. Notice that the number of real solutions for the equation can be changed by changing the value of the constant c.
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An athlete on a track team throws a shot put. The height y of the shot put in feet t seconds after it is thrown is modeled by y = –16t2 + 24.6t + 6.5. The horizontal distance x in between the athlete and the shot put is modeled by x = 29.3t. To the nearest foot, how far does the shot put land from the athlete?
Example 4: Sports Application
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Step 1 Use the first equation to determine how long it will take the shot put to hit the ground. Set the height of the shot put equal to 0 feet, and the use the quadratic formula to solve for t.
y = –16t2 + 24.6t + 6.5
Example 4 Continued
Set y equal to 0.0 = –16t2 + 24.6t + 6.5
Use the Quadratic Formula.
Substitute –16 for a, 24.6 for b, and 6.5 for c.
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Example 4 Continued
Simplify.
The time cannot be negative, so the shot put hits the ground about 1.8 seconds after it is released.
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Step 2 Find the horizontal distance that the shot put will have traveled in this time.
x = 29.3t
Example 4 Continued
Substitute 1.77 for t.
Simplify.
x ≈ 29.3(1.77)
x ≈ 51.86
x ≈ 52
The shot put will have traveled a horizontal distance of about 52 feet.
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Check Use substitution to check that the shot put hits the ground after about 1.77 seconds.
Example 4 Continued
The height is approximately equal to 0 when t = 1.77.
y = –16t2 + 24.6t + 6.5
y ≈ –16(1.77)2 + 24.6(1.77) + 6.5
y ≈ –50.13 + 43.54 + 6.5
y ≈ –0.09
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Example 4
The pilot’s altitude decreases, which changes the function describing the water’s height toy = –16t2 –2t + 400. To the nearest foot, at what horizontal distance from the target should the pilot begin releasing the water?
A pilot of a helicopter plans to release a bucket of water on a forest fire. The height y in feet of the water t seconds after its release is modeled by y = –16t2 – 2t + 500. the horizontal distance x in feet between the water and its point of release is modeled by x = 91t.
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Properties of Solving Quadratic Equations
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Properties of Solving Quadratic Equations
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No matter which method you use to solve a quadratic equation, you should get the same answer.
Helpful Hint