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Answer no 8 ,9 and 10

8) 4 + 2q²/p² - 4r/p + r²/p²9) (3/4, -9/4)10) (3/8, 41/16)

Step-by-step explanation:

8.

Given

α and β are roots of px² + qx + r = 0

The sum of the roots is α + β = -q/p, the product of then roots αβ = r/p

(2 + α²)(2 + β²) =  4 + 2(α² + β²) + (αβ)² =4 + 2((α + β)² -2αβ) + (αβ)² =4 + 2((-q/p)² - 2r/p) + (r/p)² =4 + 2q²/p² - 4r/p + r²/p²

9.

Given function

y = 2x² - 3x - 1

The minimum point is reached at vertex

The vertex is:

x = -b/2ax = -(-3)/2*2 = 3/4

The corresponding y-coordinate is:

y = 2(3/4)² - 3(3/4) - 1 = 9/8 - 9/4 - 1 = 1/8(9 - 18 - 9) = - 18/8 = - 9/4

So the point is:

(3/4, -9/4)

10.

Given function

y = 2 - 3x - 4x²

The maximum is reached at vertex

The vertex is:

x = -b/2ax = -(-3)/2(-4) = -3/8

The corresponding y-coordinate is:

y = 2 - 3(-3/8) -4(-3/8)² = 2 + 9/8 - 9/16 = 1/16(32 + 18 - 9) = 41/16

So the maximum point is:

(3/8, 41/16)

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