Algebra Examples

$- 1 = \sqrt{6 x + 21} - x$

Step 1

Rewrite the equation as $\sqrt{6 x + 21} - x = - 1$ .

$\sqrt{6 x + 21} - x = - 1$

Step 2

Add $x$ to both sides of the equation.

$\sqrt{6 x + 21} = - 1 + x$

Step 3

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

${\sqrt{6 x + 21}}^{2} = {(- 1 + x)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x + 21}$ as ${(6 x + 21)}^{\frac{1}{2}}$ .

${({(6 x + 21)}^{\frac{1}{2}})}^{2} = {(- 1 + x)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(6 x + 21)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(6 x + 21)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

${(6 x + 21)}^{\frac{1}{2} \cdot 2} = {(- 1 + x)}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

${(6 x + 21)}^{\frac{1}{2} \cdot 2} = {(- 1 + x)}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(6 x + 21)}^{1} = {(- 1 + x)}^{2}$

Step 4.2.1.2

Simplify.

$6 x + 21 = {(- 1 + x)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

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

$6 x + 21 = (- 1 + x) (- 1 + x)$

Step 4.3.1.2

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

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Step 4.3.1.2.1

Apply the distributive property.

$6 x + 21 = - 1 (- 1 + x) + x (- 1 + x)$

Step 4.3.1.2.2

Apply the distributive property.

$6 x + 21 = - 1 \cdot - 1 - 1 x + x (- 1 + x)$

Step 4.3.1.2.3

Apply the distributive property.

$6 x + 21 = - 1 \cdot - 1 - 1 x + x \cdot - 1 + x \cdot x$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Multiply $- 1$ by $- 1$ .

$6 x + 21 = 1 - 1 x + x \cdot - 1 + x \cdot x$

Step 4.3.1.3.1.2

Rewrite $- 1 x$ as $- x$ .

$6 x + 21 = 1 - x + x \cdot - 1 + x \cdot x$

Step 4.3.1.3.1.3

Move $- 1$ to the left of $x$ .

$6 x + 21 = 1 - x - 1 \cdot x + x \cdot x$

Step 4.3.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$6 x + 21 = 1 - x - x + x \cdot x$

Step 4.3.1.3.1.5

Multiply $x$ by $x$ .

$6 x + 21 = 1 - x - x + x^{2}$

Step 4.3.1.3.2

Subtract $x$ from $- x$ .

$6 x + 21 = 1 - 2 x + x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

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

$1 - 2 x + x^{2} = 6 x + 21$

Step 5.2

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

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Step 5.2.1

Subtract $6 x$ from both sides of the equation.

$1 - 2 x + x^{2} - 6 x = 21$

Step 5.2.2

Subtract $6 x$ from $- 2 x$ .

$1 + x^{2} - 8 x = 21$

Step 5.3

Subtract $21$ from both sides of the equation.

$1 + x^{2} - 8 x - 21 = 0$

Step 5.4

Subtract $21$ from $1$ .

$x^{2} - 8 x - 20 = 0$

Step 5.5

Factor $x^{2} - 8 x - 20$ using the AC method.

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Step 5.5.1

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 $- 20$ and whose sum is $- 8$ .

$- 10, 2$

Step 5.5.2

Write the factored form using these integers.

$(x - 10) (x + 2) = 0$

Step 5.6

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

$x - 10 = 0$

$x + 2 = 0$

Step 5.7

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 5.7.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 5.7.2

Add $10$ to both sides of the equation.

$x = 10$

Step 5.8

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 5.8.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 5.8.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.9

The final solution is all the values that make $(x - 10) (x + 2) = 0$ true.

$x = 10, - 2$

Step 6

Exclude the solutions that do not make $- 1 = \sqrt{6 x + 21} - x$ true.

$x = 10$