1. 3.2 Quadratic Equations, Functions, Zeros, and Models
Find zeros of quadratic functions and solve quadratic equations by using the principle of zero products, by using the principle of square roots, and by using the quadratic formula.
Solve applied problems using quadratic equations.
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Quadratic Equations
A quadratic equation is an equation that can be written in the form
ax2 + bx + c = 0, a  0,
where a, b, and c are real numbers.
A quadratic equation written in this form is said to be in standard form.
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Quadratic Functions
A quadratic function f is a function that can be written in the form
f (x) = ax2 + bx + c, a  0,
where a, b, and c are real numbers.
The zeros of a quadratic function f (x) = ax2 + bx + c are the solutions of the associated quadratic equation ax2 + bx + c = 0. Quadratic functions can have real-number or imaginary-number zeros and quadratic equations can have real-number or imaginary-number solutions.
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4. Equation-Solving Principles
The Principle of Zero Products:
If ab = 0 is true, then
a = 0 or b = 0,
and if a = 0 or b = 0,
then ab = 0.
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5. Equation-Solving Principles
The Principle of Square Roots:
If x2 = k, then
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Example
Solve 2x2  x = 3.
Solution
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7. Example - Checking the Solutions
Check: x = – 1
TRUE
The solutions are –1 and
TRUE
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Example
Solve 2x2  10 = 0.
Solution
Check:
TRUE
The solutions are and
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Quadratic Formula
The solutions of ax2 + bx + c = 0, a  0, are given by
This formula can be used to solve any quadratic equation.
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Example
Solve 3x2 + 2x = 7. Find exact solutions and approximate solutions rounded to the thousandths.
Solution:
3x2 + 2x  7 = a = 3, b = 2, c = 7
The exact solutions are:
The approximate solutions are –1.897 and
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Discriminant
When you apply the quadratic formula to any quadratic equation, you find the value of b2  4ac, which can be positive, negative, or zero This expression is called the discriminant.
For ax2 + bx + c = 0, where a, b, and c are real numbers:
b2  4ac = One real-number solution;
b2  4ac > Two different real-number solutions;
b2  4ac < Two different imaginary-number
solutions, complex conjugates.
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Applications
Some applied problems can be translated to quadratic equations.
Example
Time of Free Fall. The Petronas Towers in Kuala Lumpur, Malaysia are 1482 ft tall. How long would it take an object dropped from the top to reach the ground?
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Example (continued)
1. Familiarize. The formula s = 16t2 is used to approximate the distance s, in feet, that an object falls freely from rest in t seconds.
2. Translate. Substitute 1482 for s in the formula: = 16t2.
3. Carry out. Use the principle of square roots.
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Example (continued)
4. Check. In seconds, a dropped object would travel a distance of 16(9.624)2, or about 1482 ft. The answer checks.
5. State. It would take about sec for an object dropped from the top of the Petronas Towers to reach the ground.
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