Solve for x 4+ square root of 5x-24=x

$4 + \sqrt{5 x - 24} = x$

Subtract $4$ from both sides of the equation.

$\sqrt{5 x - 24} = x - 4$

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

${\sqrt{5 x - 24}}^{2} = {(x - 4)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 x - 24}$ as ${(5 x - 24)}^{\frac{1}{2}}$ .

${({(5 x - 24)}^{\frac{1}{2}})}^{2} = {(x - 4)}^{2}$

Simplify the left side.

Simplify ${({(5 x - 24)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 x - 24)}^{\frac{1}{2}})}^{2}$ .

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

${(5 x - 24)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 x - 24)}^{\frac{1}{2} \cdot 2} = {(x - 4)}^{2}$

Rewrite the expression.

${(5 x - 24)}^{1} = {(x - 4)}^{2}$

Simplify.

$5 x - 24 = {(x - 4)}^{2}$

Simplify the right side.

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

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

$5 x - 24 = (x - 4) (x - 4)$

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

Apply the distributive property.

$5 x - 24 = x (x - 4) - 4 (x - 4)$

Apply the distributive property.

$5 x - 24 = x \cdot x + x \cdot - 4 - 4 (x - 4)$

Apply the distributive property.

$5 x - 24 = x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$5 x - 24 = x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4$

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

$5 x - 24 = x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$5 x - 24 = x^{2} - 4 x - 4 x + 16$

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

$5 x - 24 = x^{2} - 8 x + 16$

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.

$x^{2} - 8 x + 16 = 5 x - 24$

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

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

$x^{2} - 8 x + 16 - 5 x = - 24$

Subtract $5 x$ from $- 8 x$ .

$x^{2} - 13 x + 16 = - 24$

Add $24$ to both sides of the equation.

$x^{2} - 13 x + 16 + 24 = 0$

Add $16$ and $24$ .

$x^{2} - 13 x + 40 = 0$

Factor $x^{2} - 13 x + 40$ 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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Write the factored form using these integers.

$(x - 8) (x - 5) = 0$

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

$x - 8 = 0$

$x - 5 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

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.

$x = 5$

The final solution is all the values that make $(x - 8) (x - 5) = 0$ true.

$x = 8, 5$

Solve for x 4+ square root of 5x-24=x

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