Solve by Factoring 25^(x^2)*5^(7x)=625

$25^{x^{2}} \cdot 5^{7 x} = 625$

Step 1

Subtract $625$ from both sides of the equation.

$25^{x^{2}} \cdot 5^{7 x} - 625 = 0$

Step 2

Multiply $25^{x^{2}} \cdot 5^{7 x}$ .

Rewrite $25$ as $5^{2}$ .

${(5^{2})}^{x^{2}} \cdot 5^{7 x} - 625 = 0$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5^{2 x^{2}} \cdot 5^{7 x} - 625 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{2 x^{2} + 7 x} - 625 = 0$

Step 3

Add $625$ to both sides of the equation.

$5^{2 x^{2} + 7 x} = 625$

Step 4

Create equivalent expressions in the equation that all have equal bases.

$5^{2 x^{2} + 7 x} = 5^{4}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 x^{2} + 7 x = 4$

Step 6

Solve for $x$ .

Subtract $4$ from both sides of the equation.

$2 x^{2} + 7 x - 4 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 4 = - 8$ and whose sum is $b = 7$ .

Factor $7$ out of $7 x$ .

$2 x^{2} + 7 (x) - 4 = 0$

Rewrite $7$ as $- 1$ plus $8$

$2 x^{2} + (- 1 + 8) x - 4 = 0$

Apply the distributive property.

$2 x^{2} - 1 x + 8 x - 4 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} - 1 x) + 8 x - 4 = 0$

Factor out the greatest common factor (GCF) from each group.

$x (2 x - 1) + 4 (2 x - 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$(2 x - 1) (x + 4) = 0$

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

$2 x - 1 = 0$

$x + 4 = 0$

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(2 x - 1) (x + 4) = 0$ true.

$x = \frac{1}{2}, - 4$

Solve by Factoring 25^(x^2)*5^(7x)=625

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