Finite Math Examples

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

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

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

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 2.2

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

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Step 2.2.1

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.2.2

Simplify terms.

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Step 2.2.2.1

Combine $x^{2}$ and $\frac{2}{2}$ .

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

Step 2.2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.3

Move $2$ to the left of $x^{2}$ .

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

Step 2.3

Multiply both sides by $2$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1

Cancel the common factor.

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

Step 2.4.1.1.2

Rewrite the expression.

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $7$ by $2$ .

$2 x^{2} - {(x - 1)}^{2} = 14$

Step 2.5

Solve for $x$ .

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Step 2.5.1

Subtract $14$ from both sides of the equation.

$2 x^{2} - {(x - 1)}^{2} - 14 = 0$

Step 2.5.2

Simplify $2 x^{2} - {(x - 1)}^{2} - 14$ .

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

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

$2 x^{2} - ((x - 1) (x - 1)) - 14 = 0$

Step 2.5.2.1.2

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

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Step 2.5.2.1.2.1

Apply the distributive property.

$2 x^{2} - (x (x - 1) - 1 (x - 1)) - 14 = 0$

Step 2.5.2.1.2.2

Apply the distributive property.

$2 x^{2} - (x \cdot x + x \cdot - 1 - 1 (x - 1)) - 14 = 0$

Step 2.5.2.1.2.3

Apply the distributive property.

$2 x^{2} - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) - 14 = 0$

Step 2.5.2.1.3

Simplify and combine like terms.

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Step 2.5.2.1.3.1

Simplify each term.

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Step 2.5.2.1.3.1.1

Multiply $x$ by $x$ .

$2 x^{2} - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) - 14 = 0$

Step 2.5.2.1.3.1.2

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

$2 x^{2} - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) - 14 = 0$

Step 2.5.2.1.3.1.3

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

$2 x^{2} - (x^{2} - x - 1 x - 1 \cdot - 1) - 14 = 0$

Step 2.5.2.1.3.1.4

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

$2 x^{2} - (x^{2} - x - x - 1 \cdot - 1) - 14 = 0$

Step 2.5.2.1.3.1.5

Multiply $- 1$ by $- 1$ .

$2 x^{2} - (x^{2} - x - x + 1) - 14 = 0$

Step 2.5.2.1.3.2

Subtract $x$ from $- x$ .

$2 x^{2} - (x^{2} - 2 x + 1) - 14 = 0$

Step 2.5.2.1.4

Apply the distributive property.

$2 x^{2} - x^{2} - (- 2 x) - 1 \cdot 1 - 14 = 0$

Step 2.5.2.1.5

Simplify.

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Step 2.5.2.1.5.1

Multiply $- 2$ by $- 1$ .

$2 x^{2} - x^{2} + 2 x - 1 \cdot 1 - 14 = 0$

Step 2.5.2.1.5.2

Multiply $- 1$ by $1$ .

$2 x^{2} - x^{2} + 2 x - 1 - 14 = 0$

Step 2.5.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$x^{2} + 2 x - 1 - 14 = 0$

Step 2.5.2.3

Subtract $14$ from $- 1$ .

$x^{2} + 2 x - 15 = 0$

Step 2.5.3

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 2.5.3.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 2.5.3.2

Write the factored form using these integers.

$(x - 3) (x + 5) = 0$

Step 2.5.4

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

$x - 3 = 0$

$x + 5 = 0$

Step 2.5.5

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

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Step 2.5.5.1

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

$x - 3 = 0$

Step 2.5.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.5.6

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 2.5.6.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 2.5.6.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.5.7

The final solution is all the values that make $(x - 3) (x + 5) = 0$ true.

$x = 3, - 5$

Step 2.6

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 2.7

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

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Step 2.7.1

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.7.2

Simplify terms.

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Step 2.7.2.1

Combine $x^{2}$ and $\frac{2}{2}$ .

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

Step 2.7.2.2

Combine the numerators over the common denominator.

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

Step 2.7.3

Move $2$ to the left of $x^{2}$ .

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

Step 2.8

Multiply both sides by $2$ .

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

Step 2.9

Simplify.

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Step 2.9.1

Simplify the left side.

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Step 2.9.1.1

Cancel the common factor of $2$ .

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Step 2.9.1.1.1

Cancel the common factor.

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

Step 2.9.1.1.2

Rewrite the expression.

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

Step 2.9.2

Simplify the right side.

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Step 2.9.2.1

Multiply $- 7$ by $2$ .

$2 x^{2} - {(x - 1)}^{2} = - 14$

Step 2.10

Solve for $x$ .

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Step 2.10.1

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

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Step 2.10.1.1

Add $14$ to both sides of the equation.

$2 x^{2} - {(x - 1)}^{2} + 14 = 0$

Step 2.10.1.2

Simplify $2 x^{2} - {(x - 1)}^{2} + 14$ .

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Step 2.10.1.2.1

Simplify each term.

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Step 2.10.1.2.1.1

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

$2 x^{2} - ((x - 1) (x - 1)) + 14 = 0$

Step 2.10.1.2.1.2

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

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Step 2.10.1.2.1.2.1

Apply the distributive property.

$2 x^{2} - (x (x - 1) - 1 (x - 1)) + 14 = 0$

Step 2.10.1.2.1.2.2

Apply the distributive property.

$2 x^{2} - (x \cdot x + x \cdot - 1 - 1 (x - 1)) + 14 = 0$

Step 2.10.1.2.1.2.3

Apply the distributive property.

$2 x^{2} - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + 14 = 0$

Step 2.10.1.2.1.3

Simplify and combine like terms.

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Step 2.10.1.2.1.3.1

Simplify each term.

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Step 2.10.1.2.1.3.1.1

Multiply $x$ by $x$ .

$2 x^{2} - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + 14 = 0$

Step 2.10.1.2.1.3.1.2

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

$2 x^{2} - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + 14 = 0$

Step 2.10.1.2.1.3.1.3

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

$2 x^{2} - (x^{2} - x - 1 x - 1 \cdot - 1) + 14 = 0$

Step 2.10.1.2.1.3.1.4

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

$2 x^{2} - (x^{2} - x - x - 1 \cdot - 1) + 14 = 0$

Step 2.10.1.2.1.3.1.5

Multiply $- 1$ by $- 1$ .

$2 x^{2} - (x^{2} - x - x + 1) + 14 = 0$

Step 2.10.1.2.1.3.2

Subtract $x$ from $- x$ .

$2 x^{2} - (x^{2} - 2 x + 1) + 14 = 0$

Step 2.10.1.2.1.4

Apply the distributive property.

$2 x^{2} - x^{2} - (- 2 x) - 1 \cdot 1 + 14 = 0$

Step 2.10.1.2.1.5

Simplify.

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Step 2.10.1.2.1.5.1

Multiply $- 2$ by $- 1$ .

$2 x^{2} - x^{2} + 2 x - 1 \cdot 1 + 14 = 0$

Step 2.10.1.2.1.5.2

Multiply $- 1$ by $1$ .

$2 x^{2} - x^{2} + 2 x - 1 + 14 = 0$

Step 2.10.1.2.2

Subtract $x^{2}$ from $2 x^{2}$ .

$x^{2} + 2 x - 1 + 14 = 0$

Step 2.10.1.2.3

Add $- 1$ and $14$ .

$x^{2} + 2 x + 13 = 0$

Step 2.10.2

Use the quadratic formula to find the solutions.

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

Step 2.10.3

Substitute the values $a = 1$ , $b = 2$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 13)}}{2 \cdot 1}$

Step 2.10.4

Simplify.

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Step 2.10.4.1

Simplify the numerator.

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Step 2.10.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 13}}{2 \cdot 1}$

Step 2.10.4.1.2

Multiply $- 4 \cdot 1 \cdot 13$ .

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Step 2.10.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 13}}{2 \cdot 1}$

Step 2.10.4.1.2.2

Multiply $- 4$ by $13$ .

$x = \frac{- 2 \pm \sqrt{4 - 52}}{2 \cdot 1}$

Step 2.10.4.1.3

Subtract $52$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 48}}{2 \cdot 1}$

Step 2.10.4.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 48}}{2 \cdot 1}$

Step 2.10.4.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.10.4.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{48}}{2 \cdot 1}$

Step 2.10.4.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.10.4.1.7.1

Factor $16$ out of $48$ .

$x = \frac{- 2 \pm i \cdot \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.10.4.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.10.4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (4 \sqrt{3})}{2 \cdot 1}$

Step 2.10.4.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 2 \pm 4 i \sqrt{3}}{2 \cdot 1}$

Step 2.10.4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 4 i \sqrt{3}}{2}$

Step 2.10.4.3

Simplify $\frac{- 2 \pm 4 i \sqrt{3}}{2}$ .

$x = - 1 \pm 2 i \sqrt{3}$

Step 2.10.5

The final answer is the combination of both solutions.

$x = - 1 + 2 i \sqrt{3}, - 1 - 2 i \sqrt{3}$

Step 2.11

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 5, - 1 + 2 i \sqrt{3}, - 1 - 2 i \sqrt{3}$