Find All Complex Solutions square root of 4x+29=2x-3

$\sqrt{4 x + 29} = 2 x - 3$

Step 1

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

${\sqrt{4 x + 29}}^{2} = {(2 x - 3)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x + 29}$ as ${(4 x + 29)}^{\frac{1}{2}}$ .

${({(4 x + 29)}^{\frac{1}{2}})}^{2} = {(2 x - 3)}^{2}$

Simplify the left side.

Simplify ${({(4 x + 29)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 x + 29)}^{\frac{1}{2}})}^{2}$ .

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

${(4 x + 29)}^{\frac{1}{2} \cdot 2} = {(2 x - 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 x + 29)}^{\frac{1}{2} \cdot 2} = {(2 x - 3)}^{2}$

Rewrite the expression.

${(4 x + 29)}^{1} = {(2 x - 3)}^{2}$

Simplify.

$4 x + 29 = {(2 x - 3)}^{2}$

Simplify the right side.

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

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

$4 x + 29 = (2 x - 3) (2 x - 3)$

Expand $(2 x - 3) (2 x - 3)$ using the FOIL Method.

Apply the distributive property.

$4 x + 29 = 2 x (2 x - 3) - 3 (2 x - 3)$

Apply the distributive property.

$4 x + 29 = 2 x (2 x) + 2 x \cdot - 3 - 3 (2 x - 3)$

Apply the distributive property.

$4 x + 29 = 2 x (2 x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 x + 29 = 2 \cdot 2 x \cdot x + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3$

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

Move $x$ .

$4 x + 29 = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3$

Multiply $x$ by $x$ .

$4 x + 29 = 2 \cdot 2 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3$

Multiply $2$ by $2$ .

$4 x + 29 = 4 x^{2} + 2 x \cdot - 3 - 3 (2 x) - 3 \cdot - 3$

Multiply $- 3$ by $2$ .

$4 x + 29 = 4 x^{2} - 6 x - 3 (2 x) - 3 \cdot - 3$

Multiply $2$ by $- 3$ .

$4 x + 29 = 4 x^{2} - 6 x - 6 x - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$4 x + 29 = 4 x^{2} - 6 x - 6 x + 9$

Subtract $6 x$ from $- 6 x$ .

$4 x + 29 = 4 x^{2} - 12 x + 9$

Step 3

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.

$4 x^{2} - 12 x + 9 = 4 x + 29$

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

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

$4 x^{2} - 12 x + 9 - 4 x = 29$

Subtract $4 x$ from $- 12 x$ .

$4 x^{2} - 16 x + 9 = 29$

Subtract $29$ from both sides of the equation.

$4 x^{2} - 16 x + 9 - 29 = 0$

Subtract $29$ from $9$ .

$4 x^{2} - 16 x - 20 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 x^{2} - 16 x - 20$ .

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

$4 (x^{2}) - 16 x - 20 = 0$

Factor $4$ out of $- 16 x$ .

$4 (x^{2}) + 4 (- 4 x) - 20 = 0$

Factor $4$ out of $- 20$ .

$4 x^{2} + 4 (- 4 x) + 4 \cdot - 5 = 0$

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

$4 (x^{2} - 4 x) + 4 \cdot - 5 = 0$

Factor $4$ out of $4 (x^{2} - 4 x) + 4 \cdot - 5$ .

$4 (x^{2} - 4 x - 5) = 0$

Factor.

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Write the factored form using these integers.

$4 ((x - 5) (x + 1)) = 0$

Remove unnecessary parentheses.

$4 (x - 5) (x + 1) = 0$

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

$x - 5 = 0$

$x + 1 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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 $4 (x - 5) (x + 1) = 0$ true.

$x = 5, - 1$

Step 4

Exclude the solutions that do not make $\sqrt{4 x + 29} = 2 x - 3$ true.

$x = 5$

Find All Complex Solutions square root of 4x+29=2x-3

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