Answer:
Step-by-step explanation:
Given
Total = 900
Singing = 600
Dancing = 500
None = 125
Required
Determine the number of students that participate in both
Representing; Singing with S, Dancing with D, Both singing and dancing with B and None with N;
In Set Notation;
Substitute 900 for Total, 600 for S, 500 for D and 125 for N
Open Brackets
Collect Like Terms
Collect Like Terms
Multiply both sides by -1
Reorder
<em>Hence, the number of students that do not participate in both is 325</em>
Answer:
{1,6}
Step-by-step explanation:
the second box is always where the y values/range are
F(x) = -4(x - 2)² + 2
f(x) = -4((x - 2)(x - 2)) + 2
f(x) = -4(x² - 2x - 2x + 4) + 2
f(x) = -4(x² - 4x + 4) + 2
f(x) = -4(x²) + 4(4x) - 4(4) + 2
f(x) = -4x² + 16x - 16 + 2
f(x) = -4x² + 16x - 14
-4x² + 16x - 14 = 0
x = <u>-16 +/- √(16² - 4(-4)(-14))</u>
2(-4)
x = <u>-16 +/- √(256 - 224)</u>
-8
x = <u>-16 +/- √(32)
</u> -8<u>
</u>x = <u>-16 +/- 5.66
</u> -8<u>
</u>x = <u>-16 + 5.66</u> x = <u>-16 - 5.66
</u> -8 -8<u>
</u>x = <u>-10.34</u> x = <u>-21.66</u>
-8 -8
x = 1.2925 x = 2.7075
f(x) = -4x² + 16x - 14
f(1.2925) = -4(1.2925)² + 16(1.2925) - 14
f(1,2925) = -4(1.67055625) + 20.68 - 14
f(1.2925) = -6.682225 + 20.68 - 14
f(1.2925) = 13.997775 - 14
f(1.2925) = -0.002225
(x, f(x)) = (1.2925, -0.002225)
or
f(x) = -4x² + 16x - 14
f(2.7075) = -4(2.7075)² + 16(2.7075) - 14
f(2.7075) = -4(7.33055625) + 43.32 - 14
f(2.7075) = -29.322225 + 43.32 - 14
f(2.7075) = 13.997775 - 14
f(2.7075) = -0.002225
(x, f(x)) = (2.7075, -0.002225)
--------------------------------------------------------------------------------------------
f(x) = 2(x - 2)² + 1
f(x) = 2((x - 2)(x - 2)) + 1
f(x) = 2(x² - 2x - 2x + 4) + 1
f(x) = 2(x² - 4x + 4) + 1
f(x) = 2(x²) - 2(4x) + 2(4) + 1
f(x) = 2x² - 8x + 8 + 1
f(x) = 2x² - 8x + 9
2x² - 8x + 9 = 0
x = <u>-(-8) +/- √((-8)² - 4(2)(9))
</u> <u />2(2)
x = <u>8 +/- √(64 - 72)</u>
4
x = <u>8 +/- √(-8)</u>
4
x = <u>8 +/- √(8 × (-1))</u>
4
x =<u> 8 +/- √(8)√(-1)</u>
4
x = <u>8 +/- 2.83i</u>
4
x = 2 +/- 1.415i
x = 2 + 1.415i x = 2 - 1.415i
f(x) = 2x² - 8x + 9
f(2 + 1.415i) = 2(2 + 1.415i)² - 8(2 + 1.415i) + 9
f(2 + 1.415i) = 2((2 + 1.415i)(2 + 1.415i)) - 16 - 11.32i + 9
f(2 + 1.415i) = 2(4 + 2.83i + 2.83i + 2.00225i²) - 16 - 11.32i + 9
f(2 + 1.415i) = 2(4 + 5.66i + 2.00225) - 16 - 11.32i + 9
f(2 + 1.415i) = 8 + 11.32i + 4.0045 - 16 - 11.32i + 9
f(2 + 1.415i) = 8 + 4.0045 - 16 + 9 + 11.32i - 11.32i
f(2 + 1.415i) = 12.0045 - 16 + 9
f(2 + 1.415i) = -3.9955 + 9
f(2 + 1.415i) = 5.0045
(x, f(x)) = (2 + 1.415i, 5.0045)
or
f(x) = 2x² - 8x + 9
f(2 - 1.415i) = 2(2 - 1.415i)² - 8(2 - 1.415i) + 9
f(2 - 1.415i) = 2((2 - 1.415i)(2 - 1.415i)) - 16 + 11.32i + 9
f(2 - 1.415i) = 2(4 - 2.83i - 2.83i + 2.00225i²) - 16 + 11.32i + 9
f(2 - 1.415i) = 2(4 - 5.66i + 2.00225) - 16 + 11.32i + 9
f(2 - 1.415i) = 8 - 11.32i + 4.0045 - 16 + 11.32i + 9
f(2 - 1.415i) = 8 + 4.0045 - 16 + 9 - 11.32i + 11.32i
f(2 - 1.415i) = 12.0045 - 16 + 9
f(2 - 1.145i) = -3.9955 + 9
f(2 - 1.415i) = 5.0045
(x, f(x)) = (2 - 1.415i, 5.0045)
--------------------------------------------------------------------------------------------
f(x) = -2(x - 4)² + 8
f(x) = -2((x - 4)(x - 4)) + 8
f(x) = -2(x² - 4x - 4x + 16) + 8
f(x) = -2(x² - 8x + 16) + 8
f(x) = -2(x²) + 2(8x) - 2(16) + 8
f(x) = -2x² + 16x - 32 + 8
f(x) = -2x² + 16x - 24
-2x² + 16x - 24 = 0
x = <u>-16 +/- √(16² - 4(-2)(-24))</u>
2(-2)
x = <u>-16 +/- √(256 - 192)</u>
-4
x = <u>-16 +/- √(64)</u>
-4
x = <u>-16 +/- 8</u>
-4
x = <u>-16 + 8</u> x = <u>-16 - 8</u>
-4 -4
x = <u>-8</u> x = <u>-24</u>
-4 -4
x = 2 x = 6
f(x) = -2x² + 16x - 24
f(2) = -2(2)² + 16(2) - 24
f(2) = -2(4) + 32 - 24
f(2) = -8 + 32 - 24
f(2) = 24 - 24
f(2) = 0
(x,f(x)) = (2, 0)
or
f(x) = -2x² + 16x - 24
f(6) = -2(6)² + 16(6) - 24
f(6) = -2(36) + 96 - 24
f(6) = -72 + 96 - 24
f(6) = 24 - 24
f(6) = 0
(x, f(x)) = (6, 0)
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