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8 Solve the equation 5^(2x)-6(5^(x))+5=0.

88 Solve the equation 52x−6(5x)+5=0 5^{2 x}-6\left(5^{x}\right)+5=0 .

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Q. 88 Solve the equation 52x−6(5x)+5=0 5^{2 x}-6\left(5^{x}\right)+5=0 .
  1. Substitution Simplification: Let's make a substitution to simplify the equation. Let u=5xu = 5^x.\newlineThen the equation becomes u2−6u+5=0u^2 - 6u + 5 = 0.
  2. Quadratic Equation Factoring: Now we factor the quadratic equation. \newline(u−5)(u−1)=0(u - 5)(u - 1) = 0.
  3. Solving for u: Set each factor equal to zero and solve for u.\newlineu−5=0u - 5 = 0 or u−1=0u - 1 = 0.\newlineSo, u=5u = 5 or u=1u = 1.
  4. Substitution Back to u: Now we substitute back 5x5^x for uu. \newline5x=55^x = 5 or 5x=15^x = 1.
  5. Solving for x: Solve each equation for x.\newlineFor 5x=55^x = 5, x=1x = 1 because 51=55^1 = 5.\newlineFor 5x=15^x = 1, x=0x = 0 because 50=15^0 = 1.

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