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2-2: Solving Quadratic Equations Algebraically
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:•Solve by factoring•Solve by taking square root of both sides•Solve by completing the square•Solve by using quadratic formula
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DefinitionA quadratic, or second degree equation is one that can be written in the form2 0ax bx c for real constants a,b, and c with a≠0. This is the standard form for a quadratic equation.
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Important IdeaThere are 4 techniques to algebraically solve quadratic equations:•Factoring•Taking square root of both sides•Completing the square•Using quadratic formula
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ExampleSolve by factoring:
23 10x x
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Definition
The zero product property: If the product of real numbers is zero, then one or both of the numbers must be zero
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Example
Solve by factoring:
What is wrong with this:
2 6
( 1) 6
6 & 1 6 7
x x
x x
x x x
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Try ThisSolve by factoring:
22 3 1 0t t 1
& 12
t t
Can you think of a way to check your answer?
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Try ThisSolve by factoring:
218 23 6x x
2 3 or
9 2x x
Hint: write in standard form
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ExampleSolve by taking the square root of both sides:
24 16x a.
22 15x b.
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Try ThisSolve by taking the square root of both sides:
24 16x
2x
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Try ThisSolve by taking the square root of both sides. Give exact and approximate solutions.23 16x
4 3 2.309
3x
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Example
Solve by taking the square root of both sides:
What is wrong with this?
2 4x
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ExampleComplete the square for:
2 12x x1. Half the coefficient of x: 1/2 of 12=6
2. Square this number and add to the expression
36
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Important Idea
Completing the square is the process of finding the number that will make the expression a perfect square trinomial.
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Important Idea2 12 36x x
is a perfect square trinomial because it factors as:
2( 6)( 6) ( 6)x x x
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Try This
Complete the square for: 2 8x xthen factor your result
2 28 16 ( 4)x x x
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ExampleComplete the square for: 2 3
4y y
then factor your result.Use fractions only.
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ExampleSolve by completing the square: 2 8 14 0x x 1. Move the constant to the right:
2 8 14x x
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ExampleSolve by completing the square: 2 8 14 0x x
2. Complete the square and add to the left and right:
2 8 16 14 16x x
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Example
2( 4) 2x
4 2x
Solve by completing the square: 2 8 14 0x x 3. Factor left side and solve:
4 2x
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Try ThisSolve by completing the square:
2 4 1 0x x
2 3x
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ExampleSolve by completing the square…
26 2 0x x Before you complete the square, the coefficient of the squared term must be 1
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Try ThisSolve by completing the square…fractions only.
22 13 15 0x x 3
, 52
x
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DefinitionThe solutions to
2 0ax bx c
are:2 4
2
b b acx
a
These solutions are called the Quadratic Formula
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Important Idea2 4
2
b b acx
a
2 4b ac is called thediscriminant
1. If 2 4 0b ac there are 2 real solutions
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Important Idea2 4
2
b b acx
a
2 4b ac is called thediscriminant
2. If 2 4 0b ac there is 1 real solution
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Important Idea2 4
2
b b acx
a
2 4b ac is called thediscriminant
3. If 2 4 0b ac there are no real solutions
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ExampleSolve using the quadratic formula. Leave answer in simplified radical form.
2 2 1x x
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Try ThisSolve using the quadratic formula. Leave answer in simplified radical form.24 3 5x x
3 89
8x
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ExampleSolve using the quadratic formula. Leave answer in simplified radical form.4 24 13 3 0x x
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Lesson Close
State the quadratic formula from memory.