Solve the equation without using a calculator 4^(x^2-3x+2)+4^(x^2+6x+5)=4^(2x^2+3x+7)+1

Question

Solve the equation without using a calculator


`4^(x^2-3x+2)+4^(x^2+6x+5)=4^(2x^2+3x+7)+1`

Answer 1

• x = {-5; -1; 1; 2}

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We can see that:

• The powers of the first two terms sum to the power of the third term.

Let the first term be p, second term be q, then the equation becomes:

• p + q = pq + 1
• pq - p - q + 1 = 0
• p(q - 1) - (q - 1) = 0
• (p - 1)(q - 1) = 0
• p = 1 and q = 1

Apply zero power rule:

• If aᵇ = 1 then b = 0

It gives us two equations.

1)

• x² - 3x + 2 = 0
• (x - 2)(x - 1) = 0
• x = 2 and x = 1

2)

• x² + 6x + 5 = 0
• (x + 1)(x + 5) = 0
• x = - 1 and x = - 5