Finite Math Examples

$x + 1 = \sqrt{6 x - 7} - 1$

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.

$\sqrt{6 x - 7} - 1 = x + 1$

Step 2

Move all terms not containing $\sqrt{6 x - 7}$ to the right side of the equation.

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Step 2.1

Add $1$ to both sides of the equation.

$\sqrt{6 x - 7} = x + 1 + 1$

Step 2.2

Add $1$ and $1$ .

$\sqrt{6 x - 7} = x + 2$

Step 3

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

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

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 $\sqrt{6 x - 7}$ as ${(6 x - 7)}^{\frac{1}{2}}$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(6 x - 7)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(6 x - 7)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

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

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.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

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

$6 x - 7 = (x + 2) (x + 2)$

Step 4.3.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$6 x - 7 = x (x + 2) + 2 (x + 2)$

Step 4.3.1.2.2

Apply the distributive property.

$6 x - 7 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 4.3.1.2.3

Apply the distributive property.

$6 x - 7 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

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 $x$ .

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

Step 4.3.1.3.1.2

Move $2$ to the left of $x$ .

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

Step 4.3.1.3.1.3

Multiply $2$ by $2$ .

$6 x - 7 = x^{2} + 2 x + 2 x + 4$

Step 4.3.1.3.2

Add $2 x$ and $2 x$ .

$6 x - 7 = x^{2} + 4 x + 4$

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.

$x^{2} + 4 x + 4 = 6 x - 7$

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.

$x^{2} + 4 x + 4 - 6 x = - 7$

Step 5.2.2

Subtract $6 x$ from $4 x$ .

$x^{2} - 2 x + 4 = - 7$

Step 5.3

Move all terms to the left side of the equation and simplify.

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Step 5.3.1

Add $7$ to both sides of the equation.

$x^{2} - 2 x + 4 + 7 = 0$

Step 5.3.2

Add $4$ and $7$ .

$x^{2} - 2 x + 11 = 0$

Step 5.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.5

Substitute the values $a = 1$ , $b = - 2$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 11)}}{2 \cdot 1}$

Step 5.6

Simplify.

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

Raise $- 2$ to the power of $2$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Step 5.6.1.2

Multiply $- 4 \cdot 1 \cdot 11$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 11}}{2 \cdot 1}$

Step 5.6.1.2.2

Multiply $- 4$ by $11$ .

$x = \frac{2 \pm \sqrt{4 - 44}}{2 \cdot 1}$

Step 5.6.1.3

Subtract $44$ from $4$ .

$x = \frac{2 \pm \sqrt{- 40}}{2 \cdot 1}$

Step 5.6.1.4

Rewrite $- 40$ as $- 1 (40)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 40}}{2 \cdot 1}$

Step 5.6.1.5

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{40}}{2 \cdot 1}$

Step 5.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{2 \pm i \cdot \sqrt{40}}{2 \cdot 1}$

Step 5.6.1.7

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 5.6.1.7.1

Factor $4$ out of $40$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (10)}}{2 \cdot 1}$

Step 5.6.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 5.6.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{10})}{2 \cdot 1}$

Step 5.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i \sqrt{10}}{2 \cdot 1}$

Step 5.6.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i \sqrt{10}}{2}$

Step 5.6.3

Simplify $\frac{2 \pm 2 i \sqrt{10}}{2}$ .

$x = 1 \pm i \sqrt{10}$

Step 5.7

The final answer is the combination of both solutions.

$x = 1 + i \sqrt{10}, 1 - i \sqrt{10}$