Find All Complex Solutions square root of 5x+1- square root of x=5

$\sqrt{5 x + 1} - \sqrt{x} = 5$

Step 1

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

$\sqrt{5 x + 1} = 5 + \sqrt{x}$

Step 2

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

${\sqrt{5 x + 1}}^{2} = {(5 + \sqrt{x})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 1}$ as ${(5 x + 1)}^{\frac{1}{2}}$ .

${({(5 x + 1)}^{\frac{1}{2}})}^{2} = {(5 + \sqrt{x})}^{2}$

Simplify the left side.

Simplify ${({(5 x + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x + 1)}^{\frac{1}{2}})}^{2}$ .

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

${(5 x + 1)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{x})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 x + 1)}^{\frac{1}{2} \cdot 2} = {(5 + \sqrt{x})}^{2}$

Rewrite the expression.

${(5 x + 1)}^{1} = {(5 + \sqrt{x})}^{2}$

Simplify.

$5 x + 1 = {(5 + \sqrt{x})}^{2}$

Simplify the right side.

Simplify ${(5 + \sqrt{x})}^{2}$ .

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

$5 x + 1 = (5 + \sqrt{x}) (5 + \sqrt{x})$

Expand $(5 + \sqrt{x}) (5 + \sqrt{x})$ using the FOIL Method.

Apply the distributive property.

$5 x + 1 = 5 (5 + \sqrt{x}) + \sqrt{x} (5 + \sqrt{x})$

Apply the distributive property.

$5 x + 1 = 5 \cdot 5 + 5 \sqrt{x} + \sqrt{x} (5 + \sqrt{x})$

Apply the distributive property.

$5 x + 1 = 5 \cdot 5 + 5 \sqrt{x} + \sqrt{x} \cdot 5 + \sqrt{x} \sqrt{x}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$5 x + 1 = 25 + 5 \sqrt{x} + \sqrt{x} \cdot 5 + \sqrt{x} \sqrt{x}$

Move $5$ to the left of $\sqrt{x}$ .

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \cdot \sqrt{x} + \sqrt{x} \sqrt{x}$

Multiply $\sqrt{x} \sqrt{x}$ .

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

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

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

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

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

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Add $1$ and $1$ .

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {\sqrt{x}}^{2}$

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

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

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

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

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

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

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{\frac{2}{2}}$

Rewrite the expression.

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x^{1}$

Simplify.

$5 x + 1 = 25 + 5 \sqrt{x} + 5 \sqrt{x} + x$

Add $5 \sqrt{x}$ and $5 \sqrt{x}$ .

$5 x + 1 = 25 + 10 \sqrt{x} + x$

Step 4

Solve for $10 \sqrt{x}$ .

Rewrite the equation as $25 + 10 \sqrt{x} + x = 5 x + 1$ .

$25 + 10 \sqrt{x} + x = 5 x + 1$

Move all terms not containing $10 \sqrt{x}$ to the right side of the equation.

Subtract $25$ from both sides of the equation.

$10 \sqrt{x} + x = 5 x + 1 - 25$

Subtract $x$ from both sides of the equation.

$10 \sqrt{x} = 5 x + 1 - 25 - x$

Subtract $x$ from $5 x$ .

$10 \sqrt{x} = 4 x + 1 - 25$

Subtract $25$ from $1$ .

$10 \sqrt{x} = 4 x - 24$

Step 5

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

${(10 \sqrt{x})}^{2} = {(4 x - 24)}^{2}$

Step 6

Simplify each side of the equation.

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

${(10 x^{\frac{1}{2}})}^{2} = {(4 x - 24)}^{2}$

Simplify the left side.

Simplify ${(10 x^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $10 x^{\frac{1}{2}}$ .

$10^{2} {(x^{\frac{1}{2}})}^{2} = {(4 x - 24)}^{2}$

Raise $10$ to the power of $2$ .

$100 {(x^{\frac{1}{2}})}^{2} = {(4 x - 24)}^{2}$

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

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

$100 x^{\frac{1}{2} \cdot 2} = {(4 x - 24)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$100 x^{\frac{1}{2} \cdot 2} = {(4 x - 24)}^{2}$

Rewrite the expression.

$100 x^{1} = {(4 x - 24)}^{2}$

Simplify.

$100 x = {(4 x - 24)}^{2}$

Simplify the right side.

Simplify ${(4 x - 24)}^{2}$ .

Rewrite ${(4 x - 24)}^{2}$ as $(4 x - 24) (4 x - 24)$ .

$100 x = (4 x - 24) (4 x - 24)$

Expand $(4 x - 24) (4 x - 24)$ using the FOIL Method.

Apply the distributive property.

$100 x = 4 x (4 x - 24) - 24 (4 x - 24)$

Apply the distributive property.

$100 x = 4 x (4 x) + 4 x \cdot - 24 - 24 (4 x - 24)$

Apply the distributive property.

$100 x = 4 x (4 x) + 4 x \cdot - 24 - 24 (4 x) - 24 \cdot - 24$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$100 x = 4 \cdot 4 x \cdot x + 4 x \cdot - 24 - 24 (4 x) - 24 \cdot - 24$

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

Move $x$ .

$100 x = 4 \cdot 4 (x \cdot x) + 4 x \cdot - 24 - 24 (4 x) - 24 \cdot - 24$

Multiply $x$ by $x$ .

$100 x = 4 \cdot 4 x^{2} + 4 x \cdot - 24 - 24 (4 x) - 24 \cdot - 24$

Multiply $4$ by $4$ .

$100 x = 16 x^{2} + 4 x \cdot - 24 - 24 (4 x) - 24 \cdot - 24$

Multiply $- 24$ by $4$ .

$100 x = 16 x^{2} - 96 x - 24 (4 x) - 24 \cdot - 24$

Multiply $4$ by $- 24$ .

$100 x = 16 x^{2} - 96 x - 96 x - 24 \cdot - 24$

Multiply $- 24$ by $- 24$ .

$100 x = 16 x^{2} - 96 x - 96 x + 576$

Subtract $96 x$ from $- 96 x$ .

$100 x = 16 x^{2} - 192 x + 576$

Step 7

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.

$16 x^{2} - 192 x + 576 = 100 x$

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

Subtract $100 x$ from both sides of the equation.

$16 x^{2} - 192 x + 576 - 100 x = 0$

Subtract $100 x$ from $- 192 x$ .

$16 x^{2} - 292 x + 576 = 0$

Factor the left side of the equation.

Factor $4$ out of $16 x^{2} - 292 x + 576$ .

Factor $4$ out of $16 x^{2}$ .

$4 (4 x^{2}) - 292 x + 576 = 0$

Factor $4$ out of $- 292 x$ .

$4 (4 x^{2}) + 4 (- 73 x) + 576 = 0$

Factor $4$ out of $576$ .

$4 (4 x^{2}) + 4 (- 73 x) + 4 (144) = 0$

Factor $4$ out of $4 (4 x^{2}) + 4 (- 73 x)$ .

$4 (4 x^{2} - 73 x) + 4 (144) = 0$

Factor $4$ out of $4 (4 x^{2} - 73 x) + 4 (144)$ .

$4 (4 x^{2} - 73 x + 144) = 0$

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 $a \cdot c = 4 \cdot 144 = 576$ and whose sum is $b = - 73$ .

Factor $- 73$ out of $- 73 x$ .

$4 (4 x^{2} - 73 x + 144) = 0$

Rewrite $- 73$ as $- 9$ plus $- 64$

$4 (4 x^{2} + (- 9 - 64) x + 144) = 0$

Apply the distributive property.

$4 (4 x^{2} - 9 x - 64 x + 144) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$4 ((4 x^{2} - 9 x) - 64 x + 144) = 0$

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

$4 (x (4 x - 9) - 16 (4 x - 9)) = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x - 9$ .

$4 ((4 x - 9) (x - 16)) = 0$

Remove unnecessary parentheses.

$4 (4 x - 9) (x - 16) = 0$

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

$4 x - 9 = 0$

$x - 16 = 0$

Set $4 x - 9$ equal to $0$ and solve for $x$ .

Set $4 x - 9$ equal to $0$ .

$4 x - 9 = 0$

Solve $4 x - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

$4 x = 9$

Divide each term in $4 x = 9$ by $4$ and simplify.

Divide each term in $4 x = 9$ by $4$ .

$\frac{4 x}{4} = \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{9}{4}$

Divide $x$ by $1$ .

$x = \frac{9}{4}$

Set $x - 16$ equal to $0$ and solve for $x$ .

Set $x - 16$ equal to $0$ .

$x - 16 = 0$

Add $16$ to both sides of the equation.

$x = 16$

The final solution is all the values that make $4 (4 x - 9) (x - 16) = 0$ true.

$x = \frac{9}{4}, 16$

Step 8

Exclude the solutions that do not make $\sqrt{5 x + 1} - \sqrt{x} = 5$ true.

$x = 16$

Find All Complex Solutions square root of 5x+1- square root of x=5

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