Solve by Factoring square root of 4x+13=2x-1

$\sqrt{4 x + 13} = 2 x - 1$

Step 1

Move all the expressions to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$\sqrt{4 x + 13} - 2 x = - 1$

Add $1$ to both sides of the equation.

$\sqrt{4 x + 13} - 2 x + 1 = 0$

Step 2

Move all terms not containing $\sqrt{4 x + 13}$ to the right side of the equation.

Add $2 x$ to both sides of the equation.

$\sqrt{4 x + 13} + 1 = 2 x$

Subtract $1$ from both sides of the equation.

$\sqrt{4 x + 13} = 2 x - 1$

Step 3

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

${\sqrt{4 x + 13}}^{2} = {(2 x - 1)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 13}$ as ${(4 x + 13)}^{\frac{1}{2}}$ .

${({(4 x + 13)}^{\frac{1}{2}})}^{2} = {(2 x - 1)}^{2}$

Simplify the left side.

Simplify ${({(4 x + 13)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x + 13)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x + 13)}^{\frac{1}{2} \cdot 2} = {(2 x - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x + 13)}^{\frac{1}{2} \cdot 2} = {(2 x - 1)}^{2}$

Rewrite the expression.

${(4 x + 13)}^{1} = {(2 x - 1)}^{2}$

Simplify.

$4 x + 13 = {(2 x - 1)}^{2}$

Simplify the right side.

Simplify ${(2 x - 1)}^{2}$ .

Rewrite ${(2 x - 1)}^{2}$ as $(2 x - 1) (2 x - 1)$ .

$4 x + 13 = (2 x - 1) (2 x - 1)$

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

Apply the distributive property.

$4 x + 13 = 2 x (2 x - 1) - 1 (2 x - 1)$

Apply the distributive property.

$4 x + 13 = 2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1)$

Apply the distributive property.

$4 x + 13 = 2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 x + 13 = 2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

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

Move $x$ .

$4 x + 13 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Multiply $x$ by $x$ .

$4 x + 13 = 2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Multiply $2$ by $2$ .

$4 x + 13 = 4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1$

Multiply $- 1$ by $2$ .

$4 x + 13 = 4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1$

Multiply $2$ by $- 1$ .

$4 x + 13 = 4 x^{2} - 2 x - 2 x - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$4 x + 13 = 4 x^{2} - 2 x - 2 x + 1$

Subtract $2 x$ from $- 2 x$ .

$4 x + 13 = 4 x^{2} - 4 x + 1$

Step 5

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.

$4 x^{2} - 4 x + 1 = 4 x + 13$

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

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

$4 x^{2} - 4 x + 1 - 4 x = 13$

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

$4 x^{2} - 8 x + 1 = 13$

Subtract $13$ from both sides of the equation.

$4 x^{2} - 8 x + 1 - 13 = 0$

Subtract $13$ from $1$ .

$4 x^{2} - 8 x - 12 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 8 x - 12$ .

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

$4 (x^{2}) - 8 x - 12 = 0$

Factor $4$ out of $- 8 x$ .

$4 (x^{2}) + 4 (- 2 x) - 12 = 0$

Factor $4$ out of $- 12$ .

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

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

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

Factor $4$ out of $4 (x^{2} - 2 x) + 4 \cdot - 3$ .

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

Factor.

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

$- 3, 1$

Write the factored form using these integers.

$4 ((x - 3) (x + 1)) = 0$

Remove unnecessary parentheses.

$4 (x - 3) (x + 1) = 0$

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

$x - 3 = 0$

$x + 1 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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 $4 (x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 6

Exclude the solutions that do not make $\sqrt{4 x + 13} - 2 x + 1 = 0$ true.

$x = 3$

Solve by Factoring square root of 4x+13=2x-1

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