Find the Roots (Zeros) square root of 9x+7+ square root of 2x=7

$\sqrt{9 x + 7} + \sqrt{2 x} = 7$

Subtract $\sqrt{2 x}$ from both sides of the equation.

$\sqrt{9 x + 7} = 7 - \sqrt{2 x}$

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

${\sqrt{9 x + 7}}^{2} = {(7 - \sqrt{2 x})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 x + 7}$ as ${(9 x + 7)}^{\frac{1}{2}}$ .

${({(9 x + 7)}^{\frac{1}{2}})}^{2} = {(7 - \sqrt{2 x})}^{2}$

Simplify the left side.

Simplify ${({(9 x + 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(9 x + 7)}^{\frac{1}{2}})}^{2}$ .

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

${(9 x + 7)}^{\frac{1}{2} \cdot 2} = {(7 - \sqrt{2 x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(9 x + 7)}^{\frac{1}{2} \cdot 2} = {(7 - \sqrt{2 x})}^{2}$

Rewrite the expression.

${(9 x + 7)}^{1} = {(7 - \sqrt{2 x})}^{2}$

Simplify.

$9 x + 7 = {(7 - \sqrt{2 x})}^{2}$

Simplify the right side.

Simplify ${(7 - \sqrt{2 x})}^{2}$ .

Rewrite ${(7 - \sqrt{2 x})}^{2}$ as $(7 - \sqrt{2 x}) (7 - \sqrt{2 x})$ .

$9 x + 7 = (7 - \sqrt{2 x}) (7 - \sqrt{2 x})$

Expand $(7 - \sqrt{2 x}) (7 - \sqrt{2 x})$ using the FOIL Method.

Apply the distributive property.

$9 x + 7 = 7 (7 - \sqrt{2 x}) - \sqrt{2 x} (7 - \sqrt{2 x})$

Apply the distributive property.

$9 x + 7 = 7 \cdot 7 + 7 (- \sqrt{2 x}) - \sqrt{2 x} (7 - \sqrt{2 x})$

Apply the distributive property.

$9 x + 7 = 7 \cdot 7 + 7 (- \sqrt{2 x}) - \sqrt{2 x} \cdot 7 - \sqrt{2 x} (- \sqrt{2 x})$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$9 x + 7 = 49 + 7 (- \sqrt{2 x}) - \sqrt{2 x} \cdot 7 - \sqrt{2 x} (- \sqrt{2 x})$

Multiply $- 1$ by $7$ .

$9 x + 7 = 49 - 7 \sqrt{2 x} - \sqrt{2 x} \cdot 7 - \sqrt{2 x} (- \sqrt{2 x})$

Multiply $7$ by $- 1$ .

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} - \sqrt{2 x} (- \sqrt{2 x})$

Multiply $- \sqrt{2 x} (- \sqrt{2 x})$ .

Multiply $- 1$ by $- 1$ .

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + 1 \sqrt{2 x} \sqrt{2 x}$

Multiply $\sqrt{2 x}$ by $1$ .

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + \sqrt{2 x} \sqrt{2 x}$

Raise $\sqrt{2 x}$ to the power of $1$ .

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {\sqrt{2 x}}^{1} \sqrt{2 x}$

Raise $\sqrt{2 x}$ to the power of $1$ .

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}$

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

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {\sqrt{2 x}}^{1 + 1}$

Add $1$ and $1$ .

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {\sqrt{2 x}}^{2}$

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

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

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {({(2 x)}^{\frac{1}{2}})}^{2}$

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

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {(2 x)}^{\frac{1}{2} \cdot 2}$

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

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {(2 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {(2 x)}^{\frac{2}{2}}$

Rewrite the expression.

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + {(2 x)}^{1}$

Simplify.

$9 x + 7 = 49 - 7 \sqrt{2 x} - 7 \sqrt{2 x} + 2 x$

Subtract $7 \sqrt{2 x}$ from $- 7 \sqrt{2 x}$ .

$9 x + 7 = 49 - 14 \sqrt{2 x} + 2 x$

Solve for $- 14 \sqrt{2 x}$ .

Rewrite the equation as $49 - 14 \sqrt{2 x} + 2 x = 9 x + 7$ .

$49 - 14 \sqrt{2 x} + 2 x = 9 x + 7$

Move all terms not containing $- 14 \sqrt{2 x}$ to the right side of the equation.

Subtract $49$ from both sides of the equation.

$- 14 \sqrt{2 x} + 2 x = 9 x + 7 - 49$

Subtract $2 x$ from both sides of the equation.

$- 14 \sqrt{2 x} = 9 x + 7 - 49 - 2 x$

Subtract $2 x$ from $9 x$ .

$- 14 \sqrt{2 x} = 7 x + 7 - 49$

Subtract $49$ from $7$ .

$- 14 \sqrt{2 x} = 7 x - 42$

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

${(- 14 \sqrt{2 x})}^{2} = {(7 x - 42)}^{2}$

Simplify each side of the equation.

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

${(- 14 {(2 x)}^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

Simplify the left side.

Simplify ${(- 14 {(2 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 x$ .

${(- 14 (2^{\frac{1}{2}} x^{\frac{1}{2}}))}^{2} = {(7 x - 42)}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- 14 \cdot 2^{\frac{1}{2}} x^{\frac{1}{2}}$ .

${(- 14 \cdot 2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

Apply the product rule to $- 14 \cdot 2^{\frac{1}{2}}$ .

${(- 14)}^{2} \cdot {(2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

Raise $- 14$ to the power of $2$ .

$196 \cdot {(2^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

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

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

$196 \cdot 2^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$196 \cdot 2^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

Rewrite the expression.

$196 \cdot 2^{1} {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

Evaluate the exponent.

$196 \cdot 2 {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

Multiply $196$ by $2$ .

$392 {(x^{\frac{1}{2}})}^{2} = {(7 x - 42)}^{2}$

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

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

$392 x^{\frac{1}{2} \cdot 2} = {(7 x - 42)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$392 x^{\frac{1}{2} \cdot 2} = {(7 x - 42)}^{2}$

Rewrite the expression.

$392 x^{1} = {(7 x - 42)}^{2}$

Simplify.

$392 x = {(7 x - 42)}^{2}$

Simplify the right side.

Simplify ${(7 x - 42)}^{2}$ .

Rewrite ${(7 x - 42)}^{2}$ as $(7 x - 42) (7 x - 42)$ .

$392 x = (7 x - 42) (7 x - 42)$

Expand $(7 x - 42) (7 x - 42)$ using the FOIL Method.

Apply the distributive property.

$392 x = 7 x (7 x - 42) - 42 (7 x - 42)$

Apply the distributive property.

$392 x = 7 x (7 x) + 7 x \cdot - 42 - 42 (7 x - 42)$

Apply the distributive property.

$392 x = 7 x (7 x) + 7 x \cdot - 42 - 42 (7 x) - 42 \cdot - 42$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$392 x = 7 \cdot 7 x \cdot x + 7 x \cdot - 42 - 42 (7 x) - 42 \cdot - 42$

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

Move $x$ .

$392 x = 7 \cdot 7 (x \cdot x) + 7 x \cdot - 42 - 42 (7 x) - 42 \cdot - 42$

Multiply $x$ by $x$ .

$392 x = 7 \cdot 7 x^{2} + 7 x \cdot - 42 - 42 (7 x) - 42 \cdot - 42$

Multiply $7$ by $7$ .

$392 x = 49 x^{2} + 7 x \cdot - 42 - 42 (7 x) - 42 \cdot - 42$

Multiply $- 42$ by $7$ .

$392 x = 49 x^{2} - 294 x - 42 (7 x) - 42 \cdot - 42$

Multiply $7$ by $- 42$ .

$392 x = 49 x^{2} - 294 x - 294 x - 42 \cdot - 42$

Multiply $- 42$ by $- 42$ .

$392 x = 49 x^{2} - 294 x - 294 x + 1764$

Subtract $294 x$ from $- 294 x$ .

$392 x = 49 x^{2} - 588 x + 1764$

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.

$49 x^{2} - 588 x + 1764 = 392 x$

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

Subtract $392 x$ from both sides of the equation.

$49 x^{2} - 588 x + 1764 - 392 x = 0$

Subtract $392 x$ from $- 588 x$ .

$49 x^{2} - 980 x + 1764 = 0$

Factor the left side of the equation.

Factor $49$ out of $49 x^{2} - 980 x + 1764$ .

Factor $49$ out of $49 x^{2}$ .

$49 (x^{2}) - 980 x + 1764 = 0$

Factor $49$ out of $- 980 x$ .

$49 (x^{2}) + 49 (- 20 x) + 1764 = 0$

Factor $49$ out of $1764$ .

$49 x^{2} + 49 (- 20 x) + 49 \cdot 36 = 0$

Factor $49$ out of $49 x^{2} + 49 (- 20 x)$ .

$49 (x^{2} - 20 x) + 49 \cdot 36 = 0$

Factor $49$ out of $49 (x^{2} - 20 x) + 49 \cdot 36$ .

$49 (x^{2} - 20 x + 36) = 0$

Factor.

Factor $x^{2} - 20 x + 36$ 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 $36$ and whose sum is $- 20$ .

$- 18, - 2$

Write the factored form using these integers.

$49 ((x - 18) (x - 2)) = 0$

Remove unnecessary parentheses.

$49 (x - 18) (x - 2) = 0$

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

$x - 18 = 0$

$x - 2 = 0$

Set $x - 18$ equal to $0$ and solve for $x$ .

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Add $18$ to both sides of the equation.

$x = 18$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $49 (x - 18) (x - 2) = 0$ true.

$x = 18, 2$

Exclude the solutions that do not make $\sqrt{9 x + 7} + \sqrt{2 x} = 7$ true.

$x = 2$

Find the Roots (Zeros) square root of 9x+7+ square root of 2x=7

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