Algebra 2 by Richard Wright
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Objectives:
SDA NAD Content Standards (2018): AII.4.1, AII.5.1, AII.6.3
A water fountain shoots a stream of water into the air. The height of the water can be modeled by quadratic function. Using the model to find where the water lands can be done by solving by factoring.
One of the fastest methods for solving quadratic equations is by factoring. Factoring involves writing the quadratic as the product of two factors. There are many different methods to factor quadratics, so this lesson will use a guess-and-check method.
Factoring is unmultiplying. Consider how to multiply two binomials.
(x + p)(x + q)
Distribute to get
x2 + qx + px + pq
x2 + (qx + px) + pq
Notice that the first term is product of the first terms of each binomial. Also, the last term is the product of the last terms of each binomial. Finally, the middle term is the product of the outers plus the product of the inners of the binomial.
To factor a quadratic in the form of ax2 + bx + c,
Factor x2 + 5x + 6.
Solution
Write two sets of parentheses.
( )( )
Guess the firsts. What times what makes x2? Put those in the beginning of each set of parentheses.
(x )(x )
Guess the lasts. What times what makes 6? Put those at the end of each set of parentheses.
(x + 2)(x + 3)
Check by combining the outers plus the inners and checking to see if it gives the middle term.
outers + inners = middle
3x + 2x = 5x
This is true, so the factoring is correct.
(x + 2)(x + 3)
Factor x2 − 5x − 14.
Solution
Write two sets of parentheses.
( )( )
Guess the firsts. What times what makes x2? Put those in the beginning of each set of parentheses.
(x )(x )
Guess the lasts. What times what makes −14? Put those at the end of each set of parentheses.
(x + 7)(x − 2)
Check by combining the outers plus the inners and checking to see if it gives the middle term, −5x.
outers + inners = middle
−2x + 7x = 5x
This is the correct value, but opposite sign. Fix this by switching the signs in the factors.
(x − 7)(x + 2)
Check by combining the outers plus the inners and checking to see if it gives the middle term.
outers + inners = middle
2x − 7x = −5x
This is true, so the factoring is correct.
(x − 7)(x + 2)
Factor 6x2 − 5x − 4.
Solution
Write two sets of parentheses.
( )( )
Guess the firsts. What times what makes 6x2? Put those in the beginning of each set of parentheses.
(2x )(3x )
Guess the lasts. What times what makes −4? Put those at the end of each set of parentheses.
(2x − 2)(3x + 2)
Check by combining the outers plus the inners and checking to see if it gives the middle term, −5x.
outers + inners = middle
4x − 6x = −5x
This is not true so try another combination.
Guess the firsts. What times what makes 6x2? Put those in the beginning of each set of parentheses.
(2x )(3x )
Guess the lasts. What times what makes −4? Put those at the end of each set of parentheses.
(2x + 1)(3x − 4)
Check by combining the outers plus the inners and checking to see if it gives the middle term, −5x.
outers + inners = middle
−8x + 3x = −5x
This is true, so the factoring is correct.
(2x + 1)(3x − 4)
Solving quadratic equations by factoring uses the fact that zero times anything is zero. The Zero Product Property states that if the product of two values is 0, then one or both of the values must be a zero.
If a·b = 0, then either a or b is 0.
To solve a quadratic equation by factoring,
Solve x2 + x – 20 = 0.
Solution
The equation already equals zero.
Factor the quadratic.
x2 + x – 20 = 0
(x + 5)(x – 4) = 0
Set each factor equation to zero.
x + 5 = 0 or x – 4 = 0
Solve each of those equations.
x = –5 or x = 4
Solve 3x2 – 2x – 8 = 0.
Solution
The equation already equals zero.
Factor the quadratic.
3x2 – 2x – 8 = 0
(3x + 4)(x – 2) = 0
Set each factor equation to zero.
3x + 4 = 0 or x – 2 = 0
Solve each of those equations.
3x = –4 or x = 2
x = \(\mathbf{-\frac{4}{3}}\) or x = 2
Solve 2x2 – 1 = x.
Solution
Start by making the equation equal zero.
2x2 – x – 1 = 0
Factor the quadratic.
(2x + 1)(x – 1) = 0
Set each factor equation to zero.
2x + 1 = 0 or x – 1 = 0
Solve each of those equations.
2x = –1 or x = 1
x = \(\mathbf{-\frac{1}{2}}\) or x = 1
The height of a fountain of water shot at an angle can be modeled by y = x2 – 5x where x is the distance from the fountain nozzle in feet and y is the height of the water in feet. How far away from the nozzle does the water land?
Solution
When the water lands, its height will be 0. Make y = 0 and solve for x.
0 = x2 – 5x
Factor. This does not have the constant term, so factor the common factor, x.
0 = x(x – 5)
Set each factor equal to zero. Then solve the equations.
x = 0 or x – 5 = 0
x = 0 or x = 5
The water is at zero height when it leaves the nozzle (x = 0) and when it is 5 feet away (x = 5).
95 #21, 23, 25, 27, 29, 30, 31, 36, 39, 41, 43, 45, 47, 59, 61, Mixed Review = 20
Mixed Review