Finite Math Examples

$V = {(x - a)}^{2} - x^{2}$

Step 1

Rewrite the equation as ${(x - a)}^{2} - x^{2} = V$ .

${(x - a)}^{2} - x^{2} = V$

Step 2

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

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Step 2.1

Subtract $V$ from both sides of the equation.

${(x - a)}^{2} - x^{2} - V = 0$

Step 2.2

Simplify ${(x - a)}^{2} - x^{2} - V$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$(x - a) (x - a) - x^{2} - V = 0$

Step 2.2.1.2

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

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Step 2.2.1.2.1

Apply the distributive property.

$x (x - a) - a (x - a) - x^{2} - V = 0$

Step 2.2.1.2.2

Apply the distributive property.

$x \cdot x + x (- a) - a (x - a) - x^{2} - V = 0$

Step 2.2.1.2.3

Apply the distributive property.

$x \cdot x + x (- a) - a x - a (- a) - x^{2} - V = 0$

Step 2.2.1.3

Simplify and combine like terms.

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Step 2.2.1.3.1

Simplify each term.

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Step 2.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.2.1.3.1.2

Rewrite using the commutative property of multiplication.

$x^{2} - x a - a x - a (- a) - x^{2} - V = 0$

Step 2.2.1.3.1.3

Rewrite using the commutative property of multiplication.

$x^{2} - x a - a x - 1 \cdot - 1 a \cdot a - x^{2} - V = 0$

Step 2.2.1.3.1.4

Multiply $a$ by $a$ by adding the exponents.

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Step 2.2.1.3.1.4.1

Move $a$ .

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

Step 2.2.1.3.1.4.2

Multiply $a$ by $a$ .

$x^{2} - x a - a x - 1 \cdot - 1 a^{2} - x^{2} - V = 0$

Step 2.2.1.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x a - a x + 1 a^{2} - x^{2} - V = 0$

Step 2.2.1.3.1.6

Multiply $a^{2}$ by $1$ .

$x^{2} - x a - a x + a^{2} - x^{2} - V = 0$

Step 2.2.1.3.2

Subtract $a x$ from $- x a$ .

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Step 2.2.1.3.2.1

Move $x$ .

$x^{2} - a x - a x + a^{2} - x^{2} - V = 0$

Step 2.2.1.3.2.2

Subtract $a x$ from $- a x$ .

$x^{2} - 2 a x + a^{2} - x^{2} - V = 0$

Step 2.2.2

Combine the opposite terms in $x^{2} - 2 a x + a^{2} - x^{2} - V$ .

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Step 2.2.2.1

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

$- 2 a x + a^{2} + 0 - V = 0$

Step 2.2.2.2

Add $- 2 a x + a^{2}$ and $0$ .

$- 2 a x + a^{2} - V = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Add parentheses.

$a = \frac{2 x \pm \sqrt{{(- 2 x)}^{2} - 4 \cdot (1 \cdot (- V))}}{2 \cdot 1}$

Step 5.1.2

Let $u = 1 \cdot (- V)$ . Substitute $u$ for all occurrences of $1 \cdot (- V)$ .

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Step 5.1.2.1

Apply the product rule to $- 2 x$ .

$a = \frac{2 x \pm \sqrt{{(- 2)}^{2} x^{2} - 4 \cdot u}}{2 \cdot 1}$

Step 5.1.2.2

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

$a = \frac{2 x \pm \sqrt{4 x^{2} - 4 u}}{2 \cdot 1}$

Step 5.1.3

Factor $4$ out of $4 x^{2} - 4 u$ .

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Step 5.1.3.1

Factor $4$ out of $4 x^{2}$ .

$a = \frac{2 x \pm \sqrt{4 (x^{2}) - 4 u}}{2 \cdot 1}$

Step 5.1.3.2

Factor $4$ out of $- 4 u$ .

$a = \frac{2 x \pm \sqrt{4 (x^{2}) + 4 (- u)}}{2 \cdot 1}$

Step 5.1.3.3

Factor $4$ out of $4 (x^{2}) + 4 (- u)$ .

$a = \frac{2 x \pm \sqrt{4 (x^{2} - u)}}{2 \cdot 1}$

Step 5.1.4

Replace all occurrences of $u$ with $1 \cdot (- V)$ .

$a = \frac{2 x \pm \sqrt{4 (x^{2} - (1 \cdot (- V)))}}{2 \cdot 1}$

Step 5.1.5

Simplify each term.

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Step 5.1.5.1

Multiply $- V$ by $1$ .

$a = \frac{2 x \pm \sqrt{4 (x^{2} + V)}}{2 \cdot 1}$

Step 5.1.5.2

Multiply $- - V$ .

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Step 5.1.5.2.1

Multiply $- 1$ by $- 1$ .

$a = \frac{2 x \pm \sqrt{4 (x^{2} + 1 V)}}{2 \cdot 1}$

Step 5.1.5.2.2

Multiply $V$ by $1$ .

$a = \frac{2 x \pm \sqrt{4 (x^{2} + V)}}{2 \cdot 1}$

Step 5.1.6

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

$a = \frac{2 x \pm \sqrt{2^{2} (x^{2} + V)}}{2 \cdot 1}$

Step 5.1.7

Pull terms out from under the radical.

$a = \frac{2 x \pm 2 \sqrt{x^{2} + V}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$a = \frac{2 x \pm 2 \sqrt{x^{2} + V}}{2}$

Step 5.3

Simplify $\frac{2 x \pm 2 \sqrt{x^{2} + V}}{2}$ .

$a = x \pm \sqrt{x^{2} + V}$

Step 6

The final answer is the combination of both solutions.

$a = x + \sqrt{x^{2} + V}$

$a = x - \sqrt{x^{2} + V}$