Finite Math Examples

$p^{2} \cdot k^{2} + p (k \cdot n - k) + n = 0$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$p^{2} k^{2} + p (k n) + p (- k) + n = 0$

Step 1.2

Rewrite using the commutative property of multiplication.

$p^{2} k^{2} + p k n - p k + n = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = p^{2}$ , $b = p n - p$ , and $c = n$ into the quadratic formula and solve for $k$ .

$\frac{- (p n - p) \pm \sqrt{{(p n - p)}^{2} - 4 \cdot (p^{2} \cdot n)}}{2 p^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$k = \frac{- (p n) + p \pm \sqrt{{(p n - p)}^{2} - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.2

Multiply $- - p$ .

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$k = \frac{- p n + 1 p \pm \sqrt{{(p n - p)}^{2} - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.2.2

Multiply $p$ by $1$ .

$k = \frac{- p n + p \pm \sqrt{{(p n - p)}^{2} - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.3

Rewrite ${(p n - p)}^{2}$ as $(p n - p) (p n - p)$ .

$k = \frac{- p n + p \pm \sqrt{(p n - p) (p n - p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.4

Expand $(p n - p) (p n - p)$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

$k = \frac{- p n + p \pm \sqrt{p n (p n - p) - p (p n - p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.4.2

Apply the distributive property.

$k = \frac{- p n + p \pm \sqrt{p n (p n) + p n (- p) - p (p n - p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.4.3

Apply the distributive property.

$k = \frac{- p n + p \pm \sqrt{p n (p n) + p n (- p) - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $p$ by $p$ by adding the exponents.

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Step 4.5.1.1.1

Move $p$ .

$k = \frac{- p n + p \pm \sqrt{p \cdot p n \cdot n + p n (- p) - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.1.2

Multiply $p$ by $p$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n \cdot n + p n (- p) - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.2

Multiply $n$ by $n$ by adding the exponents.

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Step 4.5.1.2.1

Move $n$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} (n \cdot n) + p n (- p) - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.2.2

Multiply $n$ by $n$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} + p n (- p) - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.3

Rewrite using the commutative property of multiplication.

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p n p - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.4

Multiply $p$ by $p$ by adding the exponents.

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Step 4.5.1.4.1

Move $p$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p \cdot p n - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.4.2

Multiply $p$ by $p$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p (p n) - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.5

Multiply $p$ by $p$ by adding the exponents.

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Step 4.5.1.5.1

Move $p$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p \cdot p n - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.5.2

Multiply $p$ by $p$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p^{2} n - p (- p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.6

Rewrite using the commutative property of multiplication.

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p^{2} n - 1 \cdot (- 1 p \cdot p) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.7

Multiply $p$ by $p$ by adding the exponents.

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Step 4.5.1.7.1

Move $p$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p^{2} n - 1 \cdot (- 1 (p \cdot p)) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.7.2

Multiply $p$ by $p$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p^{2} n - 1 \cdot (- 1 p^{2}) - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.8

Multiply $- 1$ by $- 1$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p^{2} n + 1 p^{2} - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.1.9

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

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - p^{2} n - p^{2} n + p^{2} - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.5.2

Subtract $p^{2} n$ from $- p^{2} n$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} - 2 p^{2} n + p^{2} - 4 \cdot p^{2} \cdot n}}{2 p^{2}}$

Step 4.6

Subtract $4 p^{2} n$ from $- 2 p^{2} n$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} n^{2} + p^{2} - 6 p^{2} n}}{2 p^{2}}$

Step 4.7

Rewrite $p^{2} n^{2} + p^{2} - 6 p^{2} n$ in a factored form.

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Step 4.7.1

Factor $p^{2}$ out of $p^{2} n^{2} + p^{2} - 6 p^{2} n$ .

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Step 4.7.1.1

Factor $p^{2}$ out of $p^{2} n^{2}$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} (n^{2}) + p^{2} - 6 p^{2} n}}{2 p^{2}}$

Step 4.7.1.2

Multiply by $1$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} (n^{2}) + p^{2} \cdot 1 - 6 p^{2} n}}{2 p^{2}}$

Step 4.7.1.3

Factor $p^{2}$ out of $- 6 p^{2} n$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} (n^{2}) + p^{2} \cdot 1 + p^{2} (- 6 n)}}{2 p^{2}}$

Step 4.7.1.4

Factor $p^{2}$ out of $p^{2} (n^{2}) + p^{2} \cdot 1$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} (n^{2} + 1) + p^{2} (- 6 n)}}{2 p^{2}}$

Step 4.7.1.5

Factor $p^{2}$ out of $p^{2} (n^{2} + 1) + p^{2} (- 6 n)$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} (n^{2} + 1 - 6 n)}}{2 p^{2}}$

Step 4.7.2

Reorder terms.

$k = \frac{- p n + p \pm \sqrt{p^{2} (n^{2} - 6 n + 1)}}{2 p^{2}}$

Step 4.8

Rewrite $1$ as $1^{2}$ .

$k = \frac{- p n + p \pm \sqrt{p^{2} (n^{2} - 6 n + 1^{2})}}{2 p^{2}}$

Step 4.9

Pull terms out from under the radical.

$k = \frac{- p n + p \pm p \sqrt{n^{2} - 6 n + 1^{2}}}{2 p^{2}}$

Step 4.10

One to any power is one.

$k = \frac{- p n + p \pm p \sqrt{n^{2} - 6 n + 1}}{2 p^{2}}$

Step 5

The final answer is the combination of both solutions.

$k = - \frac{n - 1 - \sqrt{n^{2} - 6 n + 1}}{2 p}$

$k = - \frac{n - 1 + \sqrt{n^{2} - 6 n + 1}}{2 p}$