Algebra Examples

$\sqrt{7 - x} = x - 1$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $x$ from both sides of the equation.

$\sqrt{7 - x} - x = - 1$

Step 1.2

Add $1$ to both sides of the equation.

$\sqrt{7 - x} - x + 1 = 0$

Step 2

Move all terms not containing $\sqrt{7 - x}$ to the right side of the equation.

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Step 2.1

Add $x$ to both sides of the equation.

$\sqrt{7 - x} + 1 = x$

Step 2.2

Subtract $1$ from both sides of the equation.

$\sqrt{7 - x} = x - 1$

Step 3

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

${\sqrt{7 - x}}^{2} = {(x - 1)}^{2}$

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 $\sqrt{7 - x}$ as ${(7 - x)}^{\frac{1}{2}}$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

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.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

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

$7 - x = (x - 1) (x - 1)$

Step 4.3.1.2

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

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Step 4.3.1.2.1

Apply the distributive property.

$7 - x = x (x - 1) - 1 (x - 1)$

Step 4.3.1.2.2

Apply the distributive property.

$7 - x = x \cdot x + x \cdot - 1 - 1 (x - 1)$

Step 4.3.1.2.3

Apply the distributive property.

$7 - x = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1$

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 $x$ by $x$ .

$7 - x = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1$

Step 4.3.1.3.1.2

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

$7 - x = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1$

Step 4.3.1.3.1.3

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

$7 - x = x^{2} - x - 1 x - 1 \cdot - 1$

Step 4.3.1.3.1.4

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

$7 - x = x^{2} - x - x - 1 \cdot - 1$

Step 4.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$7 - x = x^{2} - x - x + 1$

Step 4.3.1.3.2

Subtract $x$ from $- x$ .

$7 - x = x^{2} - 2 x + 1$

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.

$x^{2} - 2 x + 1 = 7 - x$

Step 5.2

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

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Step 5.2.1

Add $x$ to both sides of the equation.

$x^{2} - 2 x + 1 + x = 7$

Step 5.2.2

Add $- 2 x$ and $x$ .

$x^{2} - x + 1 = 7$

Step 5.3

Subtract $7$ from both sides of the equation.

$x^{2} - x + 1 - 7 = 0$

Step 5.4

Subtract $7$ from $1$ .

$x^{2} - x - 6 = 0$

Step 5.5

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 5.5.2

Write the factored form using these integers.

$(x - 3) (x + 2) = 0$

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 $0$ .

$x - 3 = 0$

$x + 2 = 0$

Step 5.7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 5.7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 5.7.2

Add $3$ to both sides of the equation.

$x = 3$

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 $(x - 3) (x + 2) = 0$ true.

$x = 3, - 2$

Step 6

Exclude the solutions that do not make $\sqrt{7 - x} - x + 1 = 0$ true.

$x = 3$