Solve for x 25^x+5^(x+1)-66=0

$25^{x} + 5^{x + 1} - 66 = 0$

Step 1

Factor the left side of the equation.

Rewrite $5^{x + 1}$ as $5^{x} \cdot 5^{1}$ .

$25^{x} + 5^{x} \cdot 5 - 66 = 0$

Rewrite $25^{x}$ as ${(5^{2})}^{x}$ .

${(5^{2})}^{x} + 5^{x} \cdot 5 - 66 = 0$

Rewrite ${(5^{2})}^{x}$ as ${(5^{x})}^{2}$ .

${(5^{x})}^{2} + 5^{x} \cdot 5 - 66 = 0$

Let $u = 5^{x}$ . Substitute $u$ for all occurrences of $5^{x}$ .

Evaluate the exponent.

$u^{2} + u \cdot 5 - 66 = 0$

Move $5$ to the left of $u$ .

$u^{2} + 5 u - 66 = 0$

Factor $u^{2} + 5 u - 66$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 66$ and whose sum is $5$ .

$- 6, 11$

Write the factored form using these integers.

$(u - 6) (u + 11) = 0$

Replace all occurrences of $u$ with $5^{x}$ .

$(5^{x} - 6) (5^{x} + 11) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5^{x} - 6 = 0$

$5^{x} + 11 = 0$

Step 3

Set $5^{x} - 6$ equal to $0$ and solve for $x$ .

Set $5^{x} - 6$ equal to $0$ .

$5^{x} - 6 = 0$

Solve $5^{x} - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$5^{x} = 6$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (5^{x}) = \ln (6)$

Expand $\ln (5^{x})$ by moving $x$ outside the logarithm.

$x \ln (5) = \ln (6)$

Divide each term in $x \ln (5) = \ln (6)$ by $\ln (5)$ and simplify.

Divide each term in $x \ln (5) = \ln (6)$ by $\ln (5)$ .

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (6)}{\ln (5)}$

Simplify the left side.

Cancel the common factor of $\ln (5)$ .

Cancel the common factor.

$\frac{x \ln (5)}{\ln (5)} = \frac{\ln (6)}{\ln (5)}$

Divide $x$ by $1$ .

$x = \frac{\ln (6)}{\ln (5)}$

Step 4

Set $5^{x} + 11$ equal to $0$ and solve for $x$ .

Set $5^{x} + 11$ equal to $0$ .

$5^{x} + 11 = 0$

Solve $5^{x} + 11 = 0$ for $x$ .

Subtract $11$ from both sides of the equation.

$5^{x} = - 11$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (5^{x}) = \ln (- 11)$

The equation cannot be solved because $\ln (- 11)$ is undefined.

Undefined

There is no solution for $5^{x} = - 11$

No solution

Step 5

The final solution is all the values that make $(5^{x} - 6) (5^{x} + 11) = 0$ true.

$x = \frac{\ln (6)}{\ln (5)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (6)}{\ln (5)}$

Decimal Form:

$x = 1.11328275 \dots$

Solve for x 25^x+5^(x+1)-66=0

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