What is x for 5^-6 = 5^(x^2 - 5x) ?

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

5^-6 = 5^(x^2-5x)

To solve this we observe that both  sides are exponents of the same base 5.

Therefore  the exponents on both sides are equal.

-6 = x^2-5x.

Add 6.

0 = x^2-5x+6

0 = x^2-3x-2x+6

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

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

x-3 = 0 or x-2 = 0

 x= 3 or x = 2.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

This is an exponential equation.

We'll write the equation again using symmetric property:

5^(x^2-5x) = 5^-6

We'll use the one to one property, because the bases are matching:

x^2 - 5x = -6

We'll add 6 both sides:

x^2 - 5x + 6 = 0

We'll apply the quadratic formula:

x1 = [5+sqrt(25 - 24)]/2

x1 = (5+1)/2

x1 = 3

x2 = (5-1)/2

x2 = 2

Verifying:

For x1 = 3:

5^(3^2-5*3) = 5^-6

5^(9-15) = 5^-6

5^-6 = 5^-6

For x2 = 2:

5^(2^2-5*2) = 5^-6

5^(4-10) = 5^-6

5^-6 = 5^-6

We'll not reject any of found solutions.

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