What is the solution for x: 3^(x^2-7x+10) - 9^3 = 0

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to solve for x given that 3^(x^2 - 7x + 10) - 9^3 = 0

3^(x^2 - 7x + 10) - 9^3 = 0

=> 3^(x^2 - 7x + 10) = 9^3

9 = 3^2, 9^3 = 3^6

=> 3^(x^2 - 7x + 10) = 3^6

as the base is the same equate the exponent

x^2 - 7x + 10 = 6

=> x^2 - 7x + 4 = 0

x1 = 7/2 + sqrt(49 - 16)/2

=> 7/2 + (sqrt 33)/2

x2 = 7/2 - (sqrt 33)/2

The values of x are 7/2 + (sqrt 33)/2 and 7/2 - (sqrt 33)/2

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll move the number 9^3 to the right side:

3^(x^2-7x+10) = 9^3

We'll create matching bases both sides. Therefore, we'll re-write 9^3 = (3^2)^3 = 3^(2*3) = 3^6

We'll re-write the equation as it follows:

3^(x^2-7x+10) = 3^6

Since the bases are matching now, we'll apply one to one rule and we'll equate the superscripts:

(x^2-7x+10)=6

We'll subtract 6 both sides:

x^2-7x+10-6=0

x^2-7x+4=0

We'll apply quadratic formula:

x1=[7+sqrt(49-16)]/2

x1=(7+sqrt33)/2

x2=(7-sqrt33)/2

The possible values of the exponent x are: {(7-sqrt33)/2 ;(7+sqrt33)/2}.

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