Download MTH 112 Section 2.3

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.
2.
3.
4.
Factoring (Zero-Product Property)
Square Root Property
Completing the Square
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
ex. x  144
2
Examples
ex. x  32  0
2
x  32
2
x  144
2
x  12
x  32
2
x  4i 2
Completing the Square
1.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
b  b  4ac
x
2a
2
Discriminant
b²- 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.