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, by completing the square, and by using the quadratic formula.
Solve equations that are quadratic in form.
Solve applied problems using quadratic equations.
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2. 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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3. 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. Example
Solve 2x2  x = 3.
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6. Equation-Solving Principles
The Principle of Square Roots: If x2 = k, then
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7. Example
Solve 2x2  10 = 0.
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8. Completing the Square
To solve a quadratic equation by completing the square:
Isolate the terms with variables on one side of the equation and arrange them in descending order.
Divide by the coefficient of the squared term if that coefficient is not 1.
Complete the square by taking half the coefficient of the first-degree term and adding its square on both sides of the equation.
Express one side of the equation as the square of a binomial.
Use the principle of square roots.
Solve for the variable.
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9. Example
Solve 2x2  1 = 3x.
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10. 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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11. Example
Solve 3x2 + 2x = 7. Find exact solutions and approximate solutions rounded to the thousandths.
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12. Quadratic in Form
Some equations can be treated as quadratic, provided that we make a suitable substitution.
Example: x4  5x2 + 4 = 0
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