4 Find the real solutions of the following equations. C 2^(x^(2-4x+6))=8 a 8^(x^(2+2x-15))=1 4^(2x^(2-11x+15))=1 d 3^(2x^(2+9x+2))=(1)/(9) e (x^2+2x-14)^5=1 f (x^2-7x+11)^8=1

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4 Find the real solutions of the following equations. C 2^(x^(2-4x+6))=8 a 8^(x^(2+2x-15))=1 4^(2x^(2-11x+15))=1 d 3^(2x^(2+9x+2))=(1)/(9) e (x^2+2x-14)^5=1 f (x^2-7x+11)^8=1

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Answer 1

Let's solve each equation step-by-step:

c) 2^{(x² - 4x + 6)} = 8

Since 8 = 2³, we can rewrite the equation as:

2^{(x² - 4x + 6)} = 2³

The bases are the same, so we can equate the exponents:

x² - 4x + 6 = 3

x² - 4x + 3 = 0

This is a quadratic equation. We can factor it:

(x - 1)(x - 3) = 0

Therefore, the real solutions are x = 1 and x = 3.

a) 8^{(x² + 2x - 15)} = 1

Since any number raised to the power of 0 equals 1, we have:

x² + 2x - 15 = 0

This is a quadratic equation. We can factor it:

(x + 5)(x - 3) = 0

Therefore, the real solutions are x = -5 and x = 3.

d) 4^{(2x² - 11x + 15)} = 1

Similar to part (a), we have:

2x² - 11x + 15 = 0

This quadratic equation can be factored:

(2x - 5)(x - 3) = 0

Therefore, the real solutions are x = 5/2 and x = 3.

d) 3^{(2x² + 9x + 2)} = 1/9

Since 1/9 = 3⁻², we can rewrite the equation as:

3^{(2x² + 9x + 2)} = 3⁻²

Equating the exponents:

2x² + 9x + 2 = -2

2x² + 9x + 4 = 0

This quadratic equation can be factored:

(2x + 1)(x + 4) = 0

Therefore, the real solutions are x = -1/2 and x = -4.

e) (x² + 2x - 14)⁵ = 1

Taking the fifth root of both sides:

x² + 2x - 14 = 1

x² + 2x - 15 = 0

(x + 5)(x - 3) = 0

Therefore, the real solutions are x = -5 and x = 3.

f) (x² - 7x + 11)⁸ = 1

Taking the eighth root of both sides:

x² - 7x + 11 = ±1

This gives us two equations:

1) x² - 7x + 11 = 1 => x² - 7x + 10 = 0 => (x - 2)(x - 5) = 0 => x = 2, x = 5

2) x² - 7x + 11 = -1 => x² - 7x + 12 = 0 => (x - 3)(x - 4) = 0 => x = 3, x = 4

Therefore, the real solutions are x = 2, x = 3, x = 4, and x = 5.

Pertanyaan

4 Find the real solutions of the following equations. C 2^(x^(2-4x+6))=8 a 8^(x^(2+2x-15))=1 4^(2x^(2-11x+15))=1 d 3^(2x^(2+9x+2))=(1)/(9) e (x^2+2x-14)^5=1 f (x^2-7x+11)^8=1

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