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Solving Quadratic Equations
What does x = ?????
Solving Quadratic EquationsWhat does x =? Five different ways:
By Graphing By Factoring By Square Root Method By Completing the Square By Quadratic Formula
Number of Solutions: There can be either 1 or 2 solutions to a
quadratic equation.
Classification of Solutions
Solutions to quadratic equations are called:“Roots” of the equation“Zeros” of the function
Solutions can be:Real (Rational or Irrational)Complex (Imaginary)
Classifying Solutions
Solutions must be in simplified radical form
If no radicals left, answers are rational.
If radical left, answers are irrational. Watch out!
If taking the square root of a negative number, answers are complex (imaginary)!!!!
By Taking Square Root
First you must isolate the x² or (x-h)² term.
Then, take the square root of both sides.
You will use ± (plus/minus) for the answer.
Examples
2
2
2
2
2 10 210
2 200
100
100
10
x
x
x
x
x
2
2
2
2
10 2 123
10 121
121
10
121
10
121
10
11 10
10
x
x
x
x
x
ix
Examples
2
2
9 169
9 169
9 13
9 13
9 13 22
9 13 4
x
x
x
x
x
x
2
2
2
7 8 625
6258
7
6258
7
6258
7
25 78
7
25 78
7
56 25 7 56 25 7,
7 7
x
x
x
x
x
x
x x
By Factoring
Place equation in standard form:ax² + bx + c = 0
Factor the expression
Use the Principle of Zero Product Rule to solve for x.
To classify: If the expression is factorable, the solutions are
“rational”• (There will be either 1 or 2 solutions)
If the expression is prime (not factorable), the solutions may be irrational or complex – not enough info to decide!
Solve Quadratic equation by factoring example Example: Put in standard form first: Factor Use principle of zero
product rule (if multiplying two things together and =0, then one of those things must be 0.)
The GCF of 4 has no relevance to final answer.
2
2
2
4 12 8
4 12 8 0
4( 3 2) 0
4( 1)( 2) 0
1 0 2 0
1 2
{ 1, 2}
r r
r r
r r
r r
r r
r r
r
By Completing the Square
Complete the square, then isolate the (x-h)² term.
Solve by square root method.
2
2
2
2
2
6 43 2
6 41
6 9 41 9
3 50
( 3) 50
3 5 2
3 5 2 3 5 2
3 5 2 3 5 2
n n
n n
n n
n
n
n
n n
n n
By Quadratic Formulaax² + bx + c = 0
2 4
2
b b acx
a
By Graphing
You have done this! Graph one side of equation in Y1, other side in Y2.
2nd Calc Intersect to find the intersection of the two functions.
Classify solutions: If graphs intersect twice, there are 2 solutions. (2 real
solutions)
If graphs intersect once, there is 1 solution (1 real solution)
If graphs never intersect, there are no “real” solutions, but there are 2 complex solutions
Discriminant-used to classify solutions of quadratic equations
The discriminant is the radicand portion of the quadratic formula: Discriminant = b²-4ac
If discriminant = 0, one rational solutionIf discriminant = perfect square number, 2
rational solutionsIf discriminant = non-perfect square number, 2
irrational solutionsIf discriminant = negative number, 2 complex
solutions
Solving Word Problems that are quadratic (area problems)
Draw a picture! Find an expression for length and width in
terms of a variable. Find an expression for area in terms of the
variable. Set the actual number for area equal to the
expression. Put quadratic equation in standard form (set
= 0) Factor and solve by factoring.
Word Problem Example
A square garden is increased by 2 on one side and decreased by 3 on the other, to form a rectangular garden. The area of the new garden is 50 m². Find the dimensions of the original garden.
The dimensions of the original squareare x by x m.
The dimensions of the new rectangle are (x + 2) by (x-3)
x m
x m
3
x + 2
Area of the new rectangle is (x + 2)(x – 3) or x² - x -6.
2
2
6 50
56 0
( 8)( 7) 0
8 0 7 0
8 7
x x
x x
x x
x x
x x
x-3
The original dimensions of the squareis 8 x 8. The new dimensions are 10 x 5
