Warm UpWarm UpWarm Up
# 1# 1# 1# 1 # 2# 2# 2# 2
2x 7# 3# 3# 3# 3
Multiply
# 4# 4# 4# 4
Multiply:
2x 5Multiply:
2x 4
Multiply
# 5# 5# 5# 5 # 6# 6# 6# 6
2x 12
Solve Solve
24x 13 5 2x 12 37
Warm UpWarm UpWarm Up
# 1# 1# 1# 1 Multiply: 2x 5
x 5 x 5 2 10x x 25 2 10x x 25
x 5
x
5 25
2x 5x
5x
Warm UpWarm UpWarm Up
# 2# 2# 2# 2 Multiply 2x 4 2 8x x 16 2 8x x 16
x 4
x
4 16
2x 4x
4x
Warm UpWarm UpWarm Up
# 3# 3# 3# 3 Multiply: 2 14x x 49 2 14x x 49 2x 7
Warm UpWarm UpWarm Up
# 4# 4# 4# 4 Multiply 2x 12 2 24x x 144 2 24x x 144
Warm UpWarm UpWarm Up
# 5# 5# 5# 5 Solve 2x 12 37
2x 25
2x 25
x 5x 5
Warm UpWarm UpWarm Up
# 6# 6# 6# 6 Solve 24x 13 5
24x 8
2x 2
2x 2
x 2x 2
Warm UpWarm UpWarm Up
2x 5 2 10x x 252 10x x 25
2x 4 2x 8x 162x 8x 16
2 14x x 49 2 14x x 49 2x 7
2x 12 2x x24 144 2x x24 144
Pg. 585, #17 – 34. Format: Free
Pg. 585, #17 – 34. Format: Free
Quadratic Equations and FunctionsQuadratic Equations and FunctionsCh 9Ch 9
Quadratic Equations and FunctionsQuadratic Equations and FunctionsCh 9Ch 9
Using Using Square RootsSquare Roots to Solve Quadratic Equations to Solve Quadratic EquationsUsing Using Square RootsSquare Roots to Solve Quadratic Equations to Solve Quadratic Equations
Solving equations in the form: 2x k c
2x xa b c 0
Exact solutions
Approx. solutions
x 7
x 2.65
x 4 13
x 7.6 and 0.4
Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations
Example 1Example 1Example 1Example 1
2x k c 2x k c
Solve: 2x 4 64 2(x 4) 64
x 4 8
x 4 x 12
x 4 8
Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations
Example 2Example 2Example 2Example 2
2x 7 23 Solve:
x 7 23
x 7 23
2(x 7) 23
x 7 23 x 7 23
Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations
Example 3Example 3Example 3Example 3 Solve
2(x 5) 81
x 5 81
x 5 81
x 5 9
x 14x 14 x 4x 4
Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations
Example 4Example 4Example 4Example 4 Solve
2(x 12) 47
x 12 47
x 12 47
x 12 47 x 12 47 x 12 47 x 12 47
Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations
Example 5Example 5Example 5Example 5 Solve
2(3x 2) 49
3x 2 7
3x 2 7
x 3x 3
5x
3
5x
3
3x 2 7 3x 2 73x 9 3x 5
Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations
Example 6Example 6Example 6Example 6 Solve
2(x 1) 16
x 1 16
Solving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic EquationsSolving Quadratic Equations
Example 7Example 7Example 7Example 7 Solve
2(x g) h
x g h
x g h
x g h x g h x g h x g h
Deriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic Formula
2x xa b c 0
2a b x x c
2 b c x
ax
a
2b
x2a
2 2
2 2
b b 4ax
2a 4a 4c
a
2 2
2
b b 4acx
2a 4a
2
2
b b 4acx
2a 4a
2b b 4acx
2a 2a
2b b 4acx
2a 2a
2b b 4acx
2a
2b b 4ac
x2a
2x xa b c y
2
2
b4a
ca
2 2
2 2
b b
4a 4a
Deriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic FormulaDeriving the Quadratic Formula
2 2
2
b b 4acx
2a 4a
2
2
b b 4acx
2a 4a
2b b 4acx
2a
2b b 4ac
x2a
2x xa b c y
Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems
2(x 13) 64 # 1# 1# 1# 1# 2# 2# 2# 2 2(x 3) 6
2(x 3) 21 # 3# 3# 3# 3# 4# 4# 4# 4 2(x 4) 36
2(x 5) 10 # 5# 5# 5# 5 # 6# 6# 6# 6 2(x 9) 144
2(2x 1) 81 # 7# 7# 7# 7 # 8# 8# 8# 8 2(x 5) 9
Solve using square roots
2(x 3) 16
Solve using square roots
2(x 3) 16
x 3 4
Take the square root of both sides of the equation(this is the inverse operation that “un-does” the exponent)
Remember to include both possibilities for the square root,i.e. the positive root and the negative root.
You now have essentially two linear equations and you’ll need to solve each to find the solutions to the original quadratic equation.
x 3 4 x 7
x 3 4 x 1
Example #1Example #1Example #1Example #1
Notice that the number 16 is a perfect square, so taking the square root of it results in a rational number
2(x 5) 13
2(x 5) 13
x 5 13
Just like example #1, you take the square root of both sides of the equation to get rid of the exponent.
Remember to include both possibilities for the square root, i.e. the positive root and the negative root (do not approximate for tonight’s homework – in other words, do not use a calculator to find the decimal representation of the irrational number.)
You now have essentially two linear equations and you’ll need to solve each to find the solutions to the original quadratic equation.
x 5 13
x 5 13
Example #2Example #2Example #2Example #2
Notice that the number 13 is NOT a perfect square, so taking the square root of it results in an irrational number.
x 5 13
x 5 13
Note: it is customary to express the solution like this: x 5 13
Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems
2(x 13) 64 # 1# 1# 1# 1# 2# 2# 2# 2 2(x 3) 6
2(x 3) 21 # 3# 3# 3# 3# 4# 4# 4# 4 2(x 4) 36
2(x 5) 10 # 5# 5# 5# 5 # 6# 6# 6# 6 2(x 9) 144
2(2x 1) 81 # 7# 7# 7# 7 # 8# 8# 8# 8 2(x 5) 9
Solve using square roots
Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems
2(x 13) 64 # 1# 1# 1# 1# 2# 2# 2# 2 2(x 3) 6
Solve using square roots
2(x 13) 64
x 13 8
x 21 x 5
x 13 8
2(x 3) 6
x 3 6
x 3 6 x 3 6
x 3 6
Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems
# 3# 3# 3# 3# 4# 4# 4# 42(x 3) 21 2(x 4) 36
Solve using square roots
2(x 3) 21
x 3 21
x 3 21 x 3 21
x 3 21
2(x 4) 36
x 4 6
x 2 x 10
x 4 6
Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems
# 5# 5# 5# 5# 6# 6# 6# 62(x 5) 10 2(x 9) 144
Solve using square roots
2(x 5) 10
x 5 10
x 5 10 x 5 10
x 5 10
2(x 9) 144
x 9 12
x 21 x 3
x 9 12
Homewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl emsHomewor k Pr obl ems
# 7# 7# 7# 7# 8# 8# 8# 82(2x 1) 81 2(x 5) 9
Solve using square roots
2(2x 1) 81
2x 1 9
x 5 x 4
2x 1 9
2(x 5) 9
2x 10 2x 8