1
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2
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Solve the system of equations by substitution.

3
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Check your solution by graphing the equation.

4
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Let's do it by substitution.

5
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We know that y is equal to this thing over here, and we

6
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also know that y is equal to this thing over here.

7
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So they're going to intersect when they both equal the same

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y, or when this thing is equal to this thing, or when

9
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negative x squared is equal to 2x squared plus 3x minus 6.

10
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The x value, which these two are equal, is going to be the

11
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x values where these y's are equal, so that's going to be

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their point of intersection.

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That will be an x and y pair that satisfies

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both of these equations.

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Let's solve for x here.

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A good starting point-- let's just add x squared to both

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sides, and we end up with 0 is equal to 3x squared

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plus 3x minus 6.

19
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Then, just to simplify this on the right hand side, we could

20
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divide both sides of the equation by 3-- we can divide

21
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everything by 3-- and we're left with 0 is equal to x

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squared plus x minus 2.

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We could do the quadratic equation, or complete the

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square, or all sorts of crazy things, but this is actually

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very factorable-- 0 is equal to two numbers, which are 2

26
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and negative 1.

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When you multiply them, you get negative 2, and when you

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add them, you get positive 1.

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So this is x plus 2 times x minus 1.

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That tells us that x plus 2 is equal to 0, or x minus 1 could

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be equal to 0.

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Subtract 2 from both sides of this.

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You could get x is equal to negative 2.

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Or add 1 to both sides of this, x is equal to 1.

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These are our two solutions-- x is equal to negative 2 or x

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is equal to 1.

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Let's verify it.

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When I put x is equal to negative 2-- let's do it over

39
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here-- what do I get?

40
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Negative 2 squared is positive 4.

41
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And then you put a negative there, so it's negative 4--

42
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the point here is negative 2, and then negative 4.

43
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That's what happens when you put negative 2 there.

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And then what happens when I put 1?

45
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1 squared is 1, and then if you put a negative there, it's

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negative 1.

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So it's 1, negative 1.

48
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And so these are both points on this equation right here,

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on this function.

50
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If you look at this one over here, if you put negative 2

51
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over here-- 2 times negative 2 squared.

52
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Negative 2 squared is positive 4.

53
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2 times positive 4 is 8.

54
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3 times negative 2 is negative 6, so 8 minus 6.

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56
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8 minus 6 is 2, and 2 minus 6 is equal to negative 4.

57
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The point negative 2, negative 4 is on this function.

58
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They both share that point in common, so

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they intersect there.

60
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If I put 1 into this, I get 2 plus 3 minus 6.

61
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2 plus 3 is 5, minus 6, when x is 1, which is negative 1.

62
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The point 1, negative 1 is also on this top graph.

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We can plot these points: negative 2 comma negative 4.

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Negative 2-- 1, 2, 3, 4-- is right there.

65
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That's going to be a point of intersection.

66
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Then we have the point 1, negative 1.

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1, negative 1 is also going to be a point of

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intersection there.

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Let's graph this second one and just verify-- they say,

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check your solution by graphing the equation.

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Let's graph the first one.

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This is pretty easy to graph-- y is equal

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to negative x squared.

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It's going to intersect the point 0, 0.

75
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It's going to be a downward slope,

76
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downward opening parabola.

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78
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When x is positive or negative 1, y is going to be negative

79
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1, because you square them and then you take the negative.

80
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When x is positive or negative 2, y is going

81
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to be negative 4.

82
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So the graph will look something like this--

83
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4, 5, 6, 7, 8, 9.

84
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The graph will look something like that.

85
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That is that equation right here.

86
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This is going to be an upward opening parabola, and a quick

87
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way-- we could do all of that completing the square, but if

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you think about it, let's just apply the quadratic equation

89
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right here.

90
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I want to show you something, a quick way of where the

91
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vertex formula comes from when you apply

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the quadratic formula.

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If you take this, and you wanted to find its zeros, you

94
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would say x is equal to negative b-- which is negative

95
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3-- plus or minus the square root of b squared-- which is

96
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9-- minus 4ac.

97
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Minus 4 times 2 times negative 6.

98
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99
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All of that over 2 times a, or 2 times 2.

100
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101
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Now, we can evaluate this and figure out the zeros of this,

102
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but the real point why I wanted to show you this is

103
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because we always have two solutions.

104
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Let me actually write down the quadratic

105
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formula here for you.

106
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The quadratic formula is negative b plus or minus the

107
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square root of b squared minus 4ac over 2a.

108
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You always end up-- or if this is a positive number-- with

109
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two solutions, and they're equal distant.

110
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They're this far away, this far over 2a away from negative

111
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b over 2a-- we could write this as negative b over 2a

112
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plus or minus the square root of b squared

113
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minus 4ac over 2a.

114
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You have at most two solutions that are equal distant from

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this x value right here.

116
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We've seen in multiple videos: what is that point that is

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equidistant from the two solutions,

118
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that is right in between?

119
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That's going to be the line of symmetry, or the x value of

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the vertex.

121
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This is where the x value of the vertex formula comes from:

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x val of vertex is negative b over 2a.

123
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If we wanted to find the vertex, the x value of this

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guy right here, we just take negative b-- which is negative

125
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3-- over 2 times a, 2 times 2, which is 4.

126
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So, x is equal to negative 3/4 is the

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vertex of this parabola.

128
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And when x is equal to negative 3/4, what is y?

129
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We could actually-- that's a little bit more complicated

130
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right here, but I'll just go through it.

131
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It's going to be 2 times 9 over 16 minus 9

132
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over 4 minus 6.

133
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Let me actually get the calculator out for this one.

134
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It's going to be 18 over 60.

135
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Let me just get the calculator out, because it will be

136
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simpler, and I don't want to waste your time doing

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arithmetic.

138
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139
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It's going to be 18 divided by 16 minus 9 divided by 4 minus

140
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6 is equal to negative 7.125.

141
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This is equal to negative 7.125.

142
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So the vertex occurs when x is negative 3/4-- when x is right

143
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there-- and y is negative 7.

144
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This is essentially 7 1/8, so 1, 2, 3, 4, 5, 6, 7.125.

145
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So it's a little over 7.

146
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The vertex of this top graph is right over there.

147
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It's symmetric around the vertex.

148
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That is the line of symmetry, and so this top graph is going

150
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to look something like this.

151
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152
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Just like that and we're done.

153
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We've found our two points of intersection, right over there

154
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and right over there, and when you graph it it

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looks pretty good.

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