L6 Quadratic Equations and the Quadratic Formula;
A quadratic equation is an equation of the form
ax 2 + bx + c = 0
where a, b, and c are real numbers and a ≠ 0 .
Solving by Factoring
If ab = 0 , then
Example: Solve by factoring.
x 2 − 10 x = −9
4 x 2 − 12 x + 9 = 0
The Square Roots Method:
The solution set to x 2 = p ( p ≥ 0) is the set of all
square roots of the number p, that is
Example. Solve by using the Square Roots Method.
x 2 = 25
( x + 2) 2 = 3
Solving Quadratic Equations by Completing the Square:
ax 2 + bx + c = 0, a≠0
1. Make sure that a = 1; if not, divide each term by a.
2. Get the constant on the right-hand side of the
3. Take 1/2 of the coefficient of x, square it, and add
this number to both sides of the equation.
4. Factor the left-hand side into a perfect square.
5. Solve for x by using the Square Roots Method.
Example: Solve by completing the square.
x2 − 8x + 3 = 0
Example: Factor by using completing the square.
x2 + 4 x − 3 =
The Quadratic Formula
The quadratic equation
ax 2 + bx + c = 0 a≠0
has the solutions (roots)
−b ± b 2 − 4ac
The quantity b 2 − 4ac , denoted D, is called the
The equation has: two distinct real roots if D > 0
one repeated real root if D = 0
no real solutions if D < 0
Note: If a < 0 , multiply both sides of the equation by a
−1 to make use of the quadratic formula easier.
The Half-coefficient Quadratic Formula:
If in the quadratic equation
ax 2 + bx + c = 0 , a≠0
b is an even number, the following formula for the roots
may be useful:
± ⎜ ⎟ − ac
2 ⎝2⎠ .
Example: Use the discriminant to determine the number
of real solutions of the quadratic equation
3x 2 − 6 x + 3 = 0
Example: Use the quadratic formula to solve
3x + x 2 − 1 = 0
3 x 2 − 2 x − 10 = 0
The Quadratic Formula and Factoring:
Theorem: If x1 and x2 are the roots of the quadratic
ax 2 + bx + c = 0 ,
then the quadratic trinomial can be factored as follows:
ax 2 + bx + c = a ( x − x1 )( x − x2 ) .
Example: Factor by using the quadratic formula
4 x2 + 5x −
Example: Find two consecutive positive integers such
that the sum of their squares is 145.
Geometry: Finding the Dimensions
The height of a triangular sign is equal to its base. The
area of the sign is 20 square feet. Find the base and the
height of the sign.
Using the position equation
s = −16t 2 + υ0t + s0 ,
find the time when an object hits the ground if it is
dropped from a building at a height of 320 feet.
Dimensions of a Field
A farmer has 380 feet of fencing to enclose two adjacent
fields. Find the dimensions that would enclose an area of
4000 square feet.
Constructing a Box
An open box is formed by cutting 1.5 inch squares from
each corner of a rectangular piece of metal whose length
is twice its width and bending up the edges. If the box is
to have a volume of 21 cubic inches, what dimensions
should the piece of metal have?