Question

Using substitution u = 5^x, or otherwise, solve 5^2x+1=7(5x)−2.

a) x=−1x=−1
b) x=0x=0
c) x=1x=1
d) x=2x=2

Answer 1

The equation 5^(2x) + 1 = 7(5^x) - 2 can be solved by substituting u = 5^x, leading to a quadratic equation. Then, we apply the quadratic formula to find the values for u, and subsequently the values for x.

Step-by-step explanation:

The equation given is 5^(2x) + 1 = 7(5^x) - 2, and we can solve it by using substitution. Let's denote u as 5^x. Substituting u into the equation gives us u^2 + 1 = 7u - 2. Now we have a quadratic equation in terms of u: u^2 - 7u + 3 = 0. To solve for u, we can use the quadratic formula, which for an equation ax^2 + bx + c = 0, is given by x = (-b ± √(b^2 - 4ac)) / (2a).

Applying this to our quadratic equation, we can find the possible values for u, and then revert back to find the values for x by taking the logarithm base 5.