Solve the Rational Equation for x square root of 3x+4=x

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

Step 1

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

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

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 4}$ as ${(3 x + 4)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x + 4)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 3

Solve for $x$ .

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

$3 x + 4 - x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $3 x + 4 - x^{2}$ .

Reorder the expression.

Move $4$ .

$3 x - x^{2} + 4 = 0$

Reorder $3 x$ and $- x^{2}$ .

$- x^{2} + 3 x + 4 = 0$

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 3 x + 4 = 0$

Factor $- 1$ out of $3 x$ .

$- (x^{2}) - (- 3 x) + 4 = 0$

Rewrite $4$ as $- 1 (- 4)$ .

$- (x^{2}) - (- 3 x) - 1 \cdot - 4 = 0$

Factor $- 1$ out of $- (x^{2}) - (- 3 x)$ .

$- (x^{2} - 3 x) - 1 \cdot - 4 = 0$

Factor $- 1$ out of $- (x^{2} - 3 x) - 1 (- 4)$ .

$- (x^{2} - 3 x - 4) = 0$

Factor.

Factor $x^{2} - 3 x - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

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

$x - 4 = 0$

$x + 1 = 0$

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

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

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

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.

$x = - 1$

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

$x = 4, - 1$

Step 4

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

$x = 4$

Solve the Rational Equation for x square root of 3x+4=x

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