Solve the Rational Equation for x square root of 5x+14=x+4

$\sqrt{5 x + 14} = x + 4$

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

${\sqrt{5 x + 14}}^{2} = {(x + 4)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x + 14}$ as ${(5 x + 14)}^{\frac{1}{2}}$ .

${({(5 x + 14)}^{\frac{1}{2}})}^{2} = {(x + 4)}^{2}$

Simplify the left side.

Simplify ${({(5 x + 14)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x + 14)}^{\frac{1}{2}})}^{2}$ .

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

${(5 x + 14)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 x + 14)}^{\frac{1}{2} \cdot 2} = {(x + 4)}^{2}$

Rewrite the expression.

${(5 x + 14)}^{1} = {(x + 4)}^{2}$

Simplify.

$5 x + 14 = {(x + 4)}^{2}$

Simplify the right side.

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

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

$5 x + 14 = (x + 4) (x + 4)$

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$5 x + 14 = x (x + 4) + 4 (x + 4)$

Apply the distributive property.

$5 x + 14 = x \cdot x + x \cdot 4 + 4 (x + 4)$

Apply the distributive property.

$5 x + 14 = x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$5 x + 14 = x^{2} + x \cdot 4 + 4 x + 4 \cdot 4$

Move $4$ to the left of $x$ .

$5 x + 14 = x^{2} + 4 \cdot x + 4 x + 4 \cdot 4$

Multiply $4$ by $4$ .

$5 x + 14 = x^{2} + 4 x + 4 x + 16$

Add $4 x$ and $4 x$ .

$5 x + 14 = x^{2} + 8 x + 16$

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.

$x^{2} + 8 x + 16 = 5 x + 14$

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

Subtract $5 x$ from both sides of the equation.

$x^{2} + 8 x + 16 - 5 x = 14$

Subtract $5 x$ from $8 x$ .

$x^{2} + 3 x + 16 = 14$

Subtract $14$ from both sides of the equation.

$x^{2} + 3 x + 16 - 14 = 0$

Subtract $14$ from $16$ .

$x^{2} + 3 x + 2 = 0$

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$(x + 1) (x + 2) = 0$

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

$x + 1 = 0$

$x + 2 = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x + 1) (x + 2) = 0$ true.

$x = - 1, - 2$

Solve the Rational Equation for x square root of 5x+14=x+4

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