Transcript Pre-Calculus Section 1.5 Equations

Pre-Calculus
Section 1.5 Equations
Objectives:
•To solve quadratics by factoring,
completing the square, and using the
quadratic formula.
•To use the discriminant to determine the
number of real solutions to a quadratic.
•To solve absolute value equations.
Quadratic Equations
The values of the variables that make an equation
true are called roots or solutions of the
equation.
A quadratic equation has the form ax2 + bx + c = 0
where a, b, and c are real numbers and a ≠ 0.
Ex 1. Solve by factoring.
a) x 2  5 x  24
x  5 x  24  0
2
x  8x  3  0
x 8  0
x  8
x 3  0
x3
b) 2 y 2  7 y  3  0
2
2y  7y  3  0
 y  32 y  1  0
y3 0
y  3
2 y 1  0
1
y
2
Ex 2. Solve by completing the square.
a) x 2  8 x  13  0
b) 3 x 2  12 x  6  0
Quadratic Formula
The roots of a quadratic equation ax2 + bx + c = 0
where a ≠ 0 are
 b  b  4ac
x
2a
2
Ex 3. Find all solutions of each equation.
a) 3 x 2  5 x  1  0
b) 4 x 2  12 x  9  0
Class Work
Find all real solutions. Use the indicated
method to solve.
2
1. x  8 x  12  0 by factoring
2. x  14 x  10  0 by completing the square
2
3. 2 x  7 x  5  0 by quadratic formula
2
The Discriminant
b  4ac is called the discriminant of a
2
quadratic equation. It tells us how many real
solutions there are to a quadratic equation.
If D > 0, then there are 2 real solutions.
If D = 0, then there is 1 real solution.
If D < 0, then there are no real solutions.
Ex 4. Use the discriminant to determine how many
real solutions of each equation. Do not solve the
equation.
a) x 2  4 x  1  0
b) 3 x 2  6 x  9  0
c) x  2 x  1  0
2
Absolute Value Equations
Ex 5. Find all real solutions.
a) 3x  1  5
b) 7 x  5  9  11
Class Work
4. Use the discriminant to determine the
number of real solutions to the equation.
x  2x  2  0
2
5. Find all real solutions.
4 x 11  25
HW #5 p55 37-61 eoo,
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