Download 1) x2 = −11 x −10 2) 4x2 −13x = −3

Math 1014: Precalculus with Transcendentals
Ch. 1: Equations and Inequalities
Sec. 1.5: Quadratic Equations
I.
Quadratic Equation
A. Definition
x is an equation that can be written in the general form
ax + bx + c = 0, where a, b, and c are real numbers, with a ≠ 0 . A quadratic equation is
A quadratic equation in
2
a second-degree polynomial equation.
B. Methods for Solving Quadratic Equations
1. Solving Quadratic Equations by Factoring
a. Zero-Product Principle
If the product of 2 algebraic expressions is zero, then at least one of the factors
is equal to zero.
If AB = 0, then A = 0 or B = 0.
b. Steps
1) Put equation in the form,
ax 2 + bx + c = 0 .
2) Factor completely.
3) Set each factor equal to 0 and solve for x.
4) Check solutions in original equation.
c. Examples:
Solve each equation by factoring.
1)
x 2 = −11x − 10
2)
4x 2 − 13x = −3
2. Solving Quadratic Equations by the Square Root Property
a. The Square Root Property
If
u is an algebraic expression and d is a nonzero number, then u 2 = d has exactly
two solutions:
If u 2 = d, then u = ± d .
b. Steps
1) Put equation in the form,
u2 = d .
2) Solve
3) Check solutions in original equation.
c. Examples:
Solve each equation by the square root property.
1)
3x 2 − 1 = 47
2)
(x − 3)2 − 36 = 0
3. Solving Quadratic Equations by Completing the Square
a. Completing the Square
2
⎛ b⎞
If x + bx is a binomial then adding ⎜ ⎟ will result in a perfect square trinomial.
⎝ 2⎠
2
2
2
b⎞
⎛ b⎞
⎛
x + bx + ⎜ ⎟ = ⎜ x + ⎟ .
⎝ 2⎠
⎝
2⎠
2
b. Steps
1) Put equation in the form,
x 2 + bx = c .
2
⎛ b⎞
2) Add ⎜ ⎟ to both sides of the equation.
⎝ 2⎠
2
3) Factor left hand side of equation,
b⎞
⎛
⎜⎝ x + ⎟⎠ and simplify right hand side.
2
4) Take square root of both sides. Make sure to take
5) Solve each resulting equation for x.
6) Check solutions in original equation.
c. Examples:
Solve each equation by completing the square.
1)
x 2 + 4x = 12
2)
x 2 − 3x − 5 = 0
± roots of RHS.
4. Solving Quadratic Equations by Quadratic Formula
a. The Quadratic Formula
The solution of a quadratic equation in general form
a ≠ 0 , are given by the quadratic formula:
ax 2 + bx + +c = 0 , with
−b ± b 2 − 4ac
x=
2a
b.
Number of Solutions
1) The value of the discriminant, b
2
− 4ac , determines the number and kind of
solutions.
b 2 − 4ac > 0 : Two unequal real solutions
2
b) b − 4ac = 0 : One real solution
2
c) b − 4ac < 0 : No real solutions; Two complex solutions
a)
c. Examples:
Solve each equation using the quadratic formula.
1)
x 2 − 6x + 9 = 0
2)
3x 2 − 4x = 4
3)
x 2 − 2x = −17
4)
5x 2 + x − 2 = 0