Section 2.3

Quadratic Equations, Functions, and Models

Quadratic Equations- second degree equations of a single variable (highest power of variable is 2)

Quadratic Equations can have at most 2 real solutions.

Quadratic Equation

Quadratic Equation Standard Form:

ax2 + bx + c = 0

where a, b, c are real numbers and a ≠ 0.

Quadratic Function

f(x) = ax2 + bx + c

where a, b, c are real numbers and a ≠ 0.

Strategies for Solving a Quadratic Equations

1. Factoring (Zero-Product Property)2. Square Root Property3. Completing the Square4. Quadratic Formula

Zero-Product Property

• If the product of two numbers is zero (0), then one of the numbers is zero (0).

ab = 0 , where a and b are real numbers

a or b must be zero

Steps for Solving Quadratic Equations by Factoring (Zero-

Product Property)

1. Set quadratic equal to zero.– ax2 + bx + c = 0 , where a, b, c are real numbers and a

≠ 0

2. Factor.

3. Set each factor equal to zero.

4. Solve each equation for the variable.

Example of Solving a Quadratic Equation

ex. Solve for x. x2 + 2x – 15 = 0

Graph of f(x) = x² +2x - 15

Zeros of a Function• The zeros of a quadratic function

f(x) = ax2 + bx + c = 0 are the solutions of the associated quadratic equation

ax2 + bx + c = 0. (These solutions are sometimes called the roots of the equation.)

• Real number zeros (solutions) are the x-coordinates of the x-intercepts of the graph of the quadratic equation.

Zeros of a Function

When f(x) = 0, then you are finding the the zero(s) of the function.

– f(x) = 0 means y = 0– Which means we are finding the x-intercept(s)

**Zero of a function is another name for x-intercept**Zero = roots = solutions = x-intercepts

Solving Quadratic Equations with the Square Root Property

x2 = k

Examples2

2

. 144

144

12

ex x

x

x

2

2

2

. 32 0

32

32

4 2

ex x

x

x

x i

Completing the Square1.Isolate the terms with variables on one side of

the equation and arrange them in descending order.

2. Divide by the coefficient of the squared term if that coefficient is not 1.

3. Complete the square by taking half the coefficient of the first-degree term and adding its square on both sides of the equation.

4. Express one side of the equation as the square of a binomial.

5. Use the principle of square roots.6. Solve for the variable.

Steps for Solving Quadratic Equations by Using Quadratic Formula

Quadratic Equation

ax2 + bx + c = 0 , where a, b, c are real numbers and a ≠ 0

Quadratic Formula

2 42

b b acx

a

Discriminantb²- 4ac

• If the value of the discriminant is positive, then there are 2 real solutions.

• If the value of the discriminant is zero, then there is 1 real solution.

• If the value of the discriminant is negative, then there are 2 imaginary solutions.