The problem provides a table of values for the relation $y = px^2 - 5x + q$. We need to find the values of $p$ and $q$, and then complete the table.

AlgebraQuadratic EquationsSystems of EquationsFunction Evaluation
2025/4/20

1. Problem Description

The problem provides a table of values for the relation y=px25x+qy = px^2 - 5x + q. We need to find the values of pp and qq, and then complete the table.

2. Solution Steps

(a) (i) Finding the values of pp and qq:
We can use the given data points from the table to form a system of equations to solve for pp and qq. We will use the points (0,12)(0, -12) and (4,0)(4, 0).
Using the point (0,12)(0, -12), we substitute x=0x=0 and y=12y=-12 into the equation y=px25x+qy = px^2 - 5x + q:
12=p(0)25(0)+q-12 = p(0)^2 - 5(0) + q
12=00+q-12 = 0 - 0 + q
q=12q = -12
Now we have y=px25x12y = px^2 - 5x - 12.
Using the point (4,0)(4, 0), we substitute x=4x=4 and y=0y=0:
0=p(4)25(4)120 = p(4)^2 - 5(4) - 12
0=16p20120 = 16p - 20 - 12
0=16p320 = 16p - 32
16p=3216p = 32
p=3216p = \frac{32}{16}
p=2p = 2
Thus, p=2p = 2 and q=12q = -12. The equation is y=2x25x12y = 2x^2 - 5x - 12.
(a) (ii) Completing the table:
Now that we have the equation y=2x25x12y = 2x^2 - 5x - 12, we can complete the table by substituting the given values of xx:
For x=1x = -1:
y=2(1)25(1)12=2(1)+512=2+512=712=5y = 2(-1)^2 - 5(-1) - 12 = 2(1) + 5 - 12 = 2 + 5 - 12 = 7 - 12 = -5
For x=2x = -2:
y=2(2)25(2)12=2(4)+1012=8+1012=1812=6y = 2(-2)^2 - 5(-2) - 12 = 2(4) + 10 - 12 = 8 + 10 - 12 = 18 - 12 = 6 (Given)
For x=3x = -3:
y=2(3)25(3)12=2(9)+1512=18+1512=3312=21y = 2(-3)^2 - 5(-3) - 12 = 2(9) + 15 - 12 = 18 + 15 - 12 = 33 - 12 = 21 (Given)
For x=1x = 1:
y=2(1)25(1)12=2(1)512=2512=312=15y = 2(1)^2 - 5(1) - 12 = 2(1) - 5 - 12 = 2 - 5 - 12 = -3 - 12 = -15
For x=2x = 2:
y=2(2)25(2)12=2(4)1012=81012=212=14y = 2(2)^2 - 5(2) - 12 = 2(4) - 10 - 12 = 8 - 10 - 12 = -2 - 12 = -14
For x=3x = 3:
y=2(3)25(3)12=2(9)1512=181512=312=9y = 2(3)^2 - 5(3) - 12 = 2(9) - 15 - 12 = 18 - 15 - 12 = 3 - 12 = -9
For x=5x = 5:
y=2(5)25(5)12=2(25)2512=502512=2512=13y = 2(5)^2 - 5(5) - 12 = 2(25) - 25 - 12 = 50 - 25 - 12 = 25 - 12 = 13 (Given)
So, the completed table is:
xx | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5
------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | --------
yy | 21 | 6 | -5 | -12 | -15 | -14 | -9 | 0 | 13

3. Final Answer

p=2p=2 and q=12q=-12.
The completed table is:
xx | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5
------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | --------
yy | 21 | 6 | -5 | -12 | -15 | -14 | -9 | 0 | 13

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