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Solve the equation 4y^2=15+7y and hence solve the equation 4(9^x)=15+7(3^x)

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Sqdancefanidn

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Answer:

  • y = -5/4, 3
  • x = 1

Step-by-step explanation:

a) The equation written in standard form is ...

  4y^2 -7y -15 = 0

We find the two factors of (4)(-15) that have a sum of -7 to be -12 and +5. Then the equation can be written and factored as ...

  4y^2 -12y +5y -15 = 0

  4y(y -3) +5(y -3) = 0

  (4y +5)(y -3) = 0

The solutions to this equation are the values of y that make the factors zero:

  4y +5 = 0   ⇒   y = -5/4

  y -3 = 0   ⇒   y = 3

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b) Let y = 3^x. Then the equation transforms to ...

  4((3^x)^2) = 15 +7(3^x)

  4y^2 = 15 +7y . . . . . . . . . same equation as above, with the same solutions

Then the value of x is ...

  log(y) = x·log(3)

  x = log(y)/log(3) . . . . . . . . y > 0

The only useful solution of the ones found above is y = 3.

  x = log(3)/log(3)

  x = 1

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