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Factorize 2a^(n+1)-7a^n-30a^(n-1)

find the range of values of k that equation x^2+(k -2)x+1=0 has real roots

Simultaneous equation y=x^2-7x+2,y=x-(5k+4) have only one set of real solution and k is constant.find k

更新:

1.Factorize 2a^(n+1)-7a^n-30a^(n-1)

find the range of values of k that equation x^2+(k -2)x+1=0 has real roots

Simultaneous equation y=x^2-7x+2,y=x-(5k+4) have only one set of real solution and k is constant.find k

Answer 1

最佳解答

Factorize 2a^(n+1)-7a^n-30a^(n-1)

2aⁿ⁺¹ - 7aⁿ - 30aⁿ⁻¹

= (2a²) aⁿ⁻¹ - (7a) aⁿ⁻¹ - 30 aⁿ⁻¹

= aⁿ⁻¹ (2a² - 7a - 30)

x² + (k - 2) x + 1 = 0

∵ The equation has real roots

∴ Δ ≥ 0

(k - 2)² - 4 (1) (1) ≥ 0

k² - 4k + 4 - 4 ≥ 0

k (k - 4) ≥ 0

k ≤ 0 or k ≥ 0

y = x² - 7x + 2　　　　　　　y = x - (5k + 4)

x² - 7x + 2 = x - (5k + 4)

x² - 7x + 2 - x + (5k + 4) = 0

x² - 8x + (5k + 6) = 0

∵ The equations have one set of real solution

∴ Δ = 0

(-8)² - 4 (1) (5k + 6) = 0

64 - 20k - 24 = 0

20k = 40

k = 2

7.

Since ABCD is a rectangle,

BE = DE

x + 6 = 3x - 2

3x - x = 6 + 2

2x = 8

x = 4

∴ D. 4

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