Solve for x square root of 25x- square root of 16x=3

$\sqrt{25 x} - \sqrt{16 x} = 3$

Solve for $\sqrt{25 x}$ .

Simplify each term.

Rewrite $16 x$ as ${(2^{2})}^{2} x$ .

Rewrite $16$ as $4^{2}$ .

$\sqrt{25 x} - \sqrt{4^{2} x} = 3$

Rewrite $4$ as $2^{2}$ .

$\sqrt{25 x} - \sqrt{{(2^{2})}^{2} x} = 3$

Pull terms out from under the radical.

$\sqrt{25 x} - (2^{2} \sqrt{x}) = 3$

Raise $2$ to the power of $2$ .

$\sqrt{25 x} - (4 \sqrt{x}) = 3$

Multiply $4$ by $- 1$ .

$\sqrt{25 x} - 4 \sqrt{x} = 3$

Add $4 \sqrt{x}$ to both sides of the equation.

$\sqrt{25 x} = 3 + 4 \sqrt{x}$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{25 x}}^{2} = {(3 + 4 \sqrt{x})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 x}$ as ${(25 x)}^{\frac{1}{2}}$ .

${({(25 x)}^{\frac{1}{2}})}^{2} = {(3 + 4 \sqrt{x})}^{2}$

Simplify the left side.

Simplify ${({(25 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(25 x)}^{\frac{1}{2}})}^{2}$ .

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

${(25 x)}^{\frac{1}{2} \cdot 2} = {(3 + 4 \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(25 x)}^{\frac{1}{2} \cdot 2} = {(3 + 4 \sqrt{x})}^{2}$

Rewrite the expression.

${(25 x)}^{1} = {(3 + 4 \sqrt{x})}^{2}$

Simplify.

$25 x = {(3 + 4 \sqrt{x})}^{2}$

Simplify the right side.

Simplify ${(3 + 4 \sqrt{x})}^{2}$ .

Rewrite ${(3 + 4 \sqrt{x})}^{2}$ as $(3 + 4 \sqrt{x}) (3 + 4 \sqrt{x})$ .

$25 x = (3 + 4 \sqrt{x}) (3 + 4 \sqrt{x})$

Expand $(3 + 4 \sqrt{x}) (3 + 4 \sqrt{x})$ using the FOIL Method.

Apply the distributive property.

$25 x = 3 (3 + 4 \sqrt{x}) + 4 \sqrt{x} (3 + 4 \sqrt{x})$

Apply the distributive property.

$25 x = 3 \cdot 3 + 3 (4 \sqrt{x}) + 4 \sqrt{x} (3 + 4 \sqrt{x})$

Apply the distributive property.

$25 x = 3 \cdot 3 + 3 (4 \sqrt{x}) + 4 \sqrt{x} \cdot 3 + 4 \sqrt{x} (4 \sqrt{x})$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$25 x = 9 + 3 (4 \sqrt{x}) + 4 \sqrt{x} \cdot 3 + 4 \sqrt{x} (4 \sqrt{x})$

Multiply $4$ by $3$ .

$25 x = 9 + 12 \sqrt{x} + 4 \sqrt{x} \cdot 3 + 4 \sqrt{x} (4 \sqrt{x})$

Multiply $3$ by $4$ .

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 4 \sqrt{x} (4 \sqrt{x})$

Multiply $4 \sqrt{x} (4 \sqrt{x})$ .

Multiply $4$ by $4$ .

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 \sqrt{x} \sqrt{x}$

Raise $\sqrt{x}$ to the power of $1$ .

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 ({\sqrt{x}}^{1} \sqrt{x})$

Raise $\sqrt{x}$ to the power of $1$ .

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 ({\sqrt{x}}^{1} {\sqrt{x}}^{1})$

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

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 {\sqrt{x}}^{1 + 1}$

Add $1$ and $1$ .

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 {\sqrt{x}}^{2}$

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 {(x^{\frac{1}{2}})}^{2}$

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

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 x^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 x^{\frac{2}{2}}$

Rewrite the expression.

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 x^{1}$

Simplify.

$25 x = 9 + 12 \sqrt{x} + 12 \sqrt{x} + 16 x$

Add $12 \sqrt{x}$ and $12 \sqrt{x}$ .

$25 x = 9 + 24 \sqrt{x} + 16 x$

Solve for $24 \sqrt{x}$ .

Rewrite the equation as $9 + 24 \sqrt{x} + 16 x = 25 x$ .

$9 + 24 \sqrt{x} + 16 x = 25 x$

Move all terms not containing $24 \sqrt{x}$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$24 \sqrt{x} + 16 x = 25 x - 9$

Subtract $16 x$ from both sides of the equation.

$24 \sqrt{x} = 25 x - 9 - 16 x$

Subtract $16 x$ from $25 x$ .

$24 \sqrt{x} = 9 x - 9$

To remove the radical on the left side of the equation, square both sides of the equation.

${(24 \sqrt{x})}^{2} = {(9 x - 9)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(24 x^{\frac{1}{2}})}^{2} = {(9 x - 9)}^{2}$

Simplify the left side.

Simplify ${(24 x^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $24 x^{\frac{1}{2}}$ .

$24^{2} {(x^{\frac{1}{2}})}^{2} = {(9 x - 9)}^{2}$

Raise $24$ to the power of $2$ .

$576 {(x^{\frac{1}{2}})}^{2} = {(9 x - 9)}^{2}$

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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

$576 x^{\frac{1}{2} \cdot 2} = {(9 x - 9)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$576 x^{\frac{1}{2} \cdot 2} = {(9 x - 9)}^{2}$

Rewrite the expression.

$576 x^{1} = {(9 x - 9)}^{2}$

Simplify.

$576 x = {(9 x - 9)}^{2}$

Simplify the right side.

Simplify ${(9 x - 9)}^{2}$ .

Rewrite ${(9 x - 9)}^{2}$ as $(9 x - 9) (9 x - 9)$ .

$576 x = (9 x - 9) (9 x - 9)$

Expand $(9 x - 9) (9 x - 9)$ using the FOIL Method.

Apply the distributive property.

$576 x = 9 x (9 x - 9) - 9 (9 x - 9)$

Apply the distributive property.

$576 x = 9 x (9 x) + 9 x \cdot - 9 - 9 (9 x - 9)$

Apply the distributive property.

$576 x = 9 x (9 x) + 9 x \cdot - 9 - 9 (9 x) - 9 \cdot - 9$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$576 x = 9 \cdot 9 x \cdot x + 9 x \cdot - 9 - 9 (9 x) - 9 \cdot - 9$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$576 x = 9 \cdot 9 (x \cdot x) + 9 x \cdot - 9 - 9 (9 x) - 9 \cdot - 9$

Multiply $x$ by $x$ .

$576 x = 9 \cdot 9 x^{2} + 9 x \cdot - 9 - 9 (9 x) - 9 \cdot - 9$

Multiply $9$ by $9$ .

$576 x = 81 x^{2} + 9 x \cdot - 9 - 9 (9 x) - 9 \cdot - 9$

Multiply $- 9$ by $9$ .

$576 x = 81 x^{2} - 81 x - 9 (9 x) - 9 \cdot - 9$

Multiply $9$ by $- 9$ .

$576 x = 81 x^{2} - 81 x - 81 x - 9 \cdot - 9$

Multiply $- 9$ by $- 9$ .

$576 x = 81 x^{2} - 81 x - 81 x + 81$

Subtract $81 x$ from $- 81 x$ .

$576 x = 81 x^{2} - 162 x + 81$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$81 x^{2} - 162 x + 81 = 576 x$

Move all terms containing $x$ to the left side of the equation.

Subtract $576 x$ from both sides of the equation.

$81 x^{2} - 162 x + 81 - 576 x = 0$

Subtract $576 x$ from $- 162 x$ .

$81 x^{2} - 738 x + 81 = 0$

Factor the left side of the equation.

Factor $9$ out of $81 x^{2} - 738 x + 81$ .

Factor $9$ out of $81 x^{2}$ .

$9 (9 x^{2}) - 738 x + 81 = 0$

Factor $9$ out of $- 738 x$ .

$9 (9 x^{2}) + 9 (- 82 x) + 81 = 0$

Factor $9$ out of $81$ .

$9 (9 x^{2}) + 9 (- 82 x) + 9 (9) = 0$

Factor $9$ out of $9 (9 x^{2}) + 9 (- 82 x)$ .

$9 (9 x^{2} - 82 x) + 9 (9) = 0$

Factor $9$ out of $9 (9 x^{2} - 82 x) + 9 (9)$ .

$9 (9 x^{2} - 82 x + 9) = 0$

Factor.

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 $a \cdot c = 9 \cdot 9 = 81$ and whose sum is $b = - 82$ .

Factor $- 82$ out of $- 82 x$ .

$9 (9 x^{2} - 82 x + 9) = 0$

Rewrite $- 82$ as $- 1$ plus $- 81$

$9 (9 x^{2} + (- 1 - 81) x + 9) = 0$

Apply the distributive property.

$9 (9 x^{2} - 1 x - 81 x + 9) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$9 ((9 x^{2} - 1 x) - 81 x + 9) = 0$

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

$9 (x (9 x - 1) - 9 (9 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $9 x - 1$ .

$9 ((9 x - 1) (x - 9)) = 0$

Remove unnecessary parentheses.

$9 (9 x - 1) (x - 9) = 0$

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

$9 x - 1 = 0$

$x - 9 = 0$

Set $9 x - 1$ equal to $0$ and solve for $x$ .

Set $9 x - 1$ equal to $0$ .

$9 x - 1 = 0$

Solve $9 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$9 x = 1$

Divide each term in $9 x = 1$ by $9$ and simplify.

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

The final solution is all the values that make $9 (9 x - 1) (x - 9) = 0$ true.

$x = \frac{1}{9}, 9$

Exclude the solutions that do not make $\sqrt{25 x} - \sqrt{16 x} = 3$ true.

$x = 9$

Solve for x square root of 25x- square root of 16x=3

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