Solve for x x+3 = square root of 4x+17

$x + 3 = \sqrt{4 x + 17}$

Step 1

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

$\sqrt{4 x + 17} = x + 3$

Step 2

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

${\sqrt{4 x + 17}}^{2} = {(x + 3)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 17}$ as ${(4 x + 17)}^{\frac{1}{2}}$ .

${({(4 x + 17)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Simplify the left side.

Simplify ${({(4 x + 17)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x + 17)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x + 17)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x + 17)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Rewrite the expression.

${(4 x + 17)}^{1} = {(x + 3)}^{2}$

Simplify.

$4 x + 17 = {(x + 3)}^{2}$

Simplify the right side.

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

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

$4 x + 17 = (x + 3) (x + 3)$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$4 x + 17 = x (x + 3) + 3 (x + 3)$

Apply the distributive property.

$4 x + 17 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Apply the distributive property.

$4 x + 17 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 x + 17 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Move $3$ to the left of $x$ .

$4 x + 17 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Multiply $3$ by $3$ .

$4 x + 17 = x^{2} + 3 x + 3 x + 9$

Add $3 x$ and $3 x$ .

$4 x + 17 = x^{2} + 6 x + 9$

Step 4

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.

$x^{2} + 6 x + 9 = 4 x + 17$

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

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

$x^{2} + 6 x + 9 - 4 x = 17$

Subtract $4 x$ from $6 x$ .

$x^{2} + 2 x + 9 = 17$

Subtract $17$ from both sides of the equation.

$x^{2} + 2 x + 9 - 17 = 0$

Subtract $17$ from $9$ .

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

Factor $x^{2} + 2 x - 8$ 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 $2$ .

$- 2, 4$

Write the factored form using these integers.

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

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

$x - 2 = 0$

$x + 4 = 0$

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.

$x = 2$

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 $(x - 2) (x + 4) = 0$ true.

$x = 2, - 4$

Step 5

Exclude the solutions that do not make $x + 3 = \sqrt{4 x + 17}$ true.

$x = 2$

Solve for x x+3 = square root of 4x+17

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