Solve the Rational Equation for x x = square root of 42-x

$x = \sqrt{42 - x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{42 - x} = x$

Step 2

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

${\sqrt{42 - x}}^{2} = x^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{42 - x}$ as ${(42 - x)}^{\frac{1}{2}}$ .

${({(42 - x)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(42 - x)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(42 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(42 - x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(42 - x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(42 - x)}^{1} = x^{2}$

Simplify.

$42 - x = x^{2}$

Step 4

Solve for $x$ .

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

$42 - x - x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $42 - x - x^{2}$ .

Reorder the expression.

Move $42$ .

$- x - x^{2} + 42 = 0$

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

$- x^{2} - x + 42 = 0$

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

$- (x^{2}) - x + 42 = 0$

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

$- (x^{2}) - (x) + 42 = 0$

Rewrite $42$ as $- 1 (- 42)$ .

$- (x^{2}) - (x) - 1 \cdot - 42 = 0$

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

$- (x^{2} + x) - 1 \cdot - 42 = 0$

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 42)$ .

$- (x^{2} + x - 42) = 0$

Factor.

Factor $x^{2} + x - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Write the factored form using these integers.

$- ((x - 6) (x + 7)) = 0$

Remove unnecessary parentheses.

$- (x - 6) (x + 7) = 0$

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

$x - 6 = 0$

$x + 7 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

The final solution is all the values that make $- (x - 6) (x + 7) = 0$ true.

$x = 6, - 7$

Step 5

Exclude the solutions that do not make $x = \sqrt{42 - x}$ true.

$x = 6$

Solve the Rational Equation for x x = square root of 42-x

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