Find the Roots (Zeros) square root of 3x+1- square root of x+8=1

$\sqrt{3 x + 1} - \sqrt{x + 8} = 1$

Step 1

Add $\sqrt{x + 8}$ to both sides of the equation.

$\sqrt{3 x + 1} = 1 + \sqrt{x + 8}$

Step 2

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

${\sqrt{3 x + 1}}^{2} = {(1 + \sqrt{x + 8})}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 1}$ as ${(3 x + 1)}^{\frac{1}{2}}$ .

${({(3 x + 1)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{x + 8})}^{2}$

Simplify the left side.

Simplify ${({(3 x + 1)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x + 1)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x + 1)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x + 8})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x + 1)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{x + 8})}^{2}$

Rewrite the expression.

${(3 x + 1)}^{1} = {(1 + \sqrt{x + 8})}^{2}$

Simplify.

$3 x + 1 = {(1 + \sqrt{x + 8})}^{2}$

Simplify the right side.

Simplify ${(1 + \sqrt{x + 8})}^{2}$ .

Rewrite ${(1 + \sqrt{x + 8})}^{2}$ as $(1 + \sqrt{x + 8}) (1 + \sqrt{x + 8})$ .

$3 x + 1 = (1 + \sqrt{x + 8}) (1 + \sqrt{x + 8})$

Expand $(1 + \sqrt{x + 8}) (1 + \sqrt{x + 8})$ using the FOIL Method.

Apply the distributive property.

$3 x + 1 = 1 (1 + \sqrt{x + 8}) + \sqrt{x + 8} (1 + \sqrt{x + 8})$

Apply the distributive property.

$3 x + 1 = 1 \cdot 1 + 1 \sqrt{x + 8} + \sqrt{x + 8} (1 + \sqrt{x + 8})$

Apply the distributive property.

$3 x + 1 = 1 \cdot 1 + 1 \sqrt{x + 8} + \sqrt{x + 8} \cdot 1 + \sqrt{x + 8} \sqrt{x + 8}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$3 x + 1 = 1 + 1 \sqrt{x + 8} + \sqrt{x + 8} \cdot 1 + \sqrt{x + 8} \sqrt{x + 8}$

Multiply $\sqrt{x + 8}$ by $1$ .

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} \cdot 1 + \sqrt{x + 8} \sqrt{x + 8}$

Multiply $\sqrt{x + 8}$ by $1$ .

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + \sqrt{x + 8} \sqrt{x + 8}$

Multiply $\sqrt{x + 8} \sqrt{x + 8}$ .

Raise $\sqrt{x + 8}$ to the power of $1$ .

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {\sqrt{x + 8}}^{1} \sqrt{x + 8}$

Raise $\sqrt{x + 8}$ to the power of $1$ .

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {\sqrt{x + 8}}^{1} {\sqrt{x + 8}}^{1}$

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

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {\sqrt{x + 8}}^{1 + 1}$

Add $1$ and $1$ .

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {\sqrt{x + 8}}^{2}$

Rewrite ${\sqrt{x + 8}}^{2}$ as $x + 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {({(x + 8)}^{\frac{1}{2}})}^{2}$

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

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {(x + 8)}^{\frac{1}{2} \cdot 2}$

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

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {(x + 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {(x + 8)}^{\frac{2}{2}}$

Rewrite the expression.

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + {(x + 8)}^{1}$

Simplify.

$3 x + 1 = 1 + \sqrt{x + 8} + \sqrt{x + 8} + x + 8$

Add $1$ and $8$ .

$3 x + 1 = 9 + \sqrt{x + 8} + \sqrt{x + 8} + x$

Add $\sqrt{x + 8}$ and $\sqrt{x + 8}$ .

$3 x + 1 = 9 + 2 \sqrt{x + 8} + x$

Step 4

Solve for $2 \sqrt{x + 8}$ .

Rewrite the equation as $9 + 2 \sqrt{x + 8} + x = 3 x + 1$ .

$9 + 2 \sqrt{x + 8} + x = 3 x + 1$

Move all terms not containing $2 \sqrt{x + 8}$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$2 \sqrt{x + 8} + x = 3 x + 1 - 9$

Subtract $x$ from both sides of the equation.

$2 \sqrt{x + 8} = 3 x + 1 - 9 - x$

Subtract $x$ from $3 x$ .

$2 \sqrt{x + 8} = 2 x + 1 - 9$

Subtract $9$ from $1$ .

$2 \sqrt{x + 8} = 2 x - 8$

Step 5

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

${(2 \sqrt{x + 8})}^{2} = {(2 x - 8)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 8}$ as ${(x + 8)}^{\frac{1}{2}}$ .

${(2 {(x + 8)}^{\frac{1}{2}})}^{2} = {(2 x - 8)}^{2}$

Simplify the left side.

Simplify ${(2 {(x + 8)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(x + 8)}^{\frac{1}{2}}$ .

$2^{2} {({(x + 8)}^{\frac{1}{2}})}^{2} = {(2 x - 8)}^{2}$

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

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

Multiply the exponents in ${({(x + 8)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(x + 8)}^{\frac{1}{2} \cdot 2} = {(2 x - 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x + 8)}^{\frac{1}{2} \cdot 2} = {(2 x - 8)}^{2}$

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

$4 x + 4 \cdot 8 = {(2 x - 8)}^{2}$

Multiply $4$ by $8$ .

$4 x + 32 = {(2 x - 8)}^{2}$

Simplify the right side.

Simplify ${(2 x - 8)}^{2}$ .

Rewrite ${(2 x - 8)}^{2}$ as $(2 x - 8) (2 x - 8)$ .

$4 x + 32 = (2 x - 8) (2 x - 8)$

Expand $(2 x - 8) (2 x - 8)$ using the FOIL Method.

Apply the distributive property.

$4 x + 32 = 2 x (2 x - 8) - 8 (2 x - 8)$

Apply the distributive property.

$4 x + 32 = 2 x (2 x) + 2 x \cdot - 8 - 8 (2 x - 8)$

Apply the distributive property.

$4 x + 32 = 2 x (2 x) + 2 x \cdot - 8 - 8 (2 x) - 8 \cdot - 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 x + 32 = 2 \cdot 2 x \cdot x + 2 x \cdot - 8 - 8 (2 x) - 8 \cdot - 8$

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

Move $x$ .

$4 x + 32 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 8 - 8 (2 x) - 8 \cdot - 8$

Multiply $x$ by $x$ .

$4 x + 32 = 2 \cdot 2 x^{2} + 2 x \cdot - 8 - 8 (2 x) - 8 \cdot - 8$

Multiply $2$ by $2$ .

$4 x + 32 = 4 x^{2} + 2 x \cdot - 8 - 8 (2 x) - 8 \cdot - 8$

Multiply $- 8$ by $2$ .

$4 x + 32 = 4 x^{2} - 16 x - 8 (2 x) - 8 \cdot - 8$

Multiply $2$ by $- 8$ .

$4 x + 32 = 4 x^{2} - 16 x - 16 x - 8 \cdot - 8$

Multiply $- 8$ by $- 8$ .

$4 x + 32 = 4 x^{2} - 16 x - 16 x + 64$

Subtract $16 x$ from $- 16 x$ .

$4 x + 32 = 4 x^{2} - 32 x + 64$

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.

$4 x^{2} - 32 x + 64 = 4 x + 32$

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

Subtract $4 x$ from both sides of the equation.

$4 x^{2} - 32 x + 64 - 4 x = 32$

Subtract $4 x$ from $- 32 x$ .

$4 x^{2} - 36 x + 64 = 32$

Subtract $32$ from both sides of the equation.

$4 x^{2} - 36 x + 64 - 32 = 0$

Subtract $32$ from $64$ .

$4 x^{2} - 36 x + 32 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 36 x + 32$ .

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

$4 (x^{2}) - 36 x + 32 = 0$

Factor $4$ out of $- 36 x$ .

$4 (x^{2}) + 4 (- 9 x) + 32 = 0$

Factor $4$ out of $32$ .

$4 x^{2} + 4 (- 9 x) + 4 \cdot 8 = 0$

Factor $4$ out of $4 x^{2} + 4 (- 9 x)$ .

$4 (x^{2} - 9 x) + 4 \cdot 8 = 0$

Factor $4$ out of $4 (x^{2} - 9 x) + 4 \cdot 8$ .

$4 (x^{2} - 9 x + 8) = 0$

Factor.

Factor $x^{2} - 9 x + 8$ 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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Write the factored form using these integers.

$4 ((x - 8) (x - 1)) = 0$

Remove unnecessary parentheses.

$4 (x - 8) (x - 1) = 0$

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

$x - 8 = 0$

$x - 1 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $4 (x - 8) (x - 1) = 0$ true.

$x = 8, 1$

Step 8

Exclude the solutions that do not make $\sqrt{3 x + 1} - \sqrt{x + 8} = 1$ true.

$x = 8$

Step 9

Find the Roots (Zeros) square root of 3x+1- square root of x+8=1

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