Solve the Rational Equation for x square root of 32-7x=x-6

$\sqrt{32 - 7 x} = x - 6$

Step 1

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

${\sqrt{32 - 7 x}}^{2} = {(x - 6)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{32 - 7 x}$ as ${(32 - 7 x)}^{\frac{1}{2}}$ .

${({(32 - 7 x)}^{\frac{1}{2}})}^{2} = {(x - 6)}^{2}$

Simplify the left side.

Simplify ${({(32 - 7 x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(32 - 7 x)}^{\frac{1}{2}})}^{2}$ .

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

${(32 - 7 x)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(32 - 7 x)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Rewrite the expression.

${(32 - 7 x)}^{1} = {(x - 6)}^{2}$

Simplify.

$32 - 7 x = {(x - 6)}^{2}$

Simplify the right side.

Simplify ${(x - 6)}^{2}$ .

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$32 - 7 x = (x - 6) (x - 6)$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$32 - 7 x = x (x - 6) - 6 (x - 6)$

Apply the distributive property.

$32 - 7 x = x \cdot x + x \cdot - 6 - 6 (x - 6)$

Apply the distributive property.

$32 - 7 x = x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$32 - 7 x = x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6$

Move $- 6$ to the left of $x$ .

$32 - 7 x = x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$32 - 7 x = x^{2} - 6 x - 6 x + 36$

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

$32 - 7 x = x^{2} - 12 x + 36$

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.

$x^{2} - 12 x + 36 = 32 - 7 x$

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

Add $7 x$ to both sides of the equation.

$x^{2} - 12 x + 36 + 7 x = 32$

Add $- 12 x$ and $7 x$ .

$x^{2} - 5 x + 36 = 32$

Subtract $32$ from both sides of the equation.

$x^{2} - 5 x + 36 - 32 = 0$

Subtract $32$ from $36$ .

$x^{2} - 5 x + 4 = 0$

Factor $x^{2} - 5 x + 4$ 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 $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

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

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 $0$ .

$x - 1 = 0$

Add $1$ to 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 $\sqrt{32 - 7 x} = x - 6$ true.

$No solution$

Solve the Rational Equation for x square root of 32-7x=x-6

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