Finite Math Examples

$\frac{d}{49} + \frac{d^{2}}{340^{2}} = 1225$

Step 1

Raise $340$ to the power of $2$ .

$\frac{d}{49} + \frac{d^{2}}{115600} = 1225$

Step 2

Subtract $1225$ from both sides of the equation.

$\frac{d}{49} + \frac{d^{2}}{115600} - 1225 = 0$

Step 3

Multiply through by the least common denominator $5664400$ , then simplify.

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Step 3.1

Apply the distributive property.

$5664400 (\frac{d}{49}) + 5664400 (\frac{d^{2}}{115600}) + 5664400 \cdot - 1225 = 0$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $49$ .

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Step 3.2.1.1

Factor $49$ out of $5664400$ .

$49 (115600) (\frac{d}{49}) + 5664400 (\frac{d^{2}}{115600}) + 5664400 \cdot - 1225 = 0$

Step 3.2.1.2

Cancel the common factor.

$49 \cdot (115600 (\frac{d}{49})) + 5664400 (\frac{d^{2}}{115600}) + 5664400 \cdot - 1225 = 0$

Step 3.2.1.3

Rewrite the expression.

$115600 d + 5664400 (\frac{d^{2}}{115600}) + 5664400 \cdot - 1225 = 0$

Step 3.2.2

Cancel the common factor of $115600$ .

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Step 3.2.2.1

Factor $115600$ out of $5664400$ .

$115600 d + 115600 (49) (\frac{d^{2}}{115600}) + 5664400 \cdot - 1225 = 0$

Step 3.2.2.2

Cancel the common factor.

$115600 d + 115600 \cdot (49 (\frac{d^{2}}{115600})) + 5664400 \cdot - 1225 = 0$

Step 3.2.2.3

Rewrite the expression.

$115600 d + 49 d^{2} + 5664400 \cdot - 1225 = 0$

Step 3.2.3

Multiply $5664400$ by $- 1225$ .

$115600 d + 49 d^{2} - 6938890000 = 0$

Step 3.3

Reorder $115600 d$ and $49 d^{2}$ .

$49 d^{2} + 115600 d - 6938890000 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 49$ , $b = 115600$ , and $c = - 6938890000$ into the quadratic formula and solve for $d$ .

$\frac{- 115600 \pm \sqrt{115600^{2} - 4 \cdot (49 \cdot - 6938890000)}}{2 \cdot 49}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $115600$ to the power of $2$ .

$d = \frac{- 115600 \pm \sqrt{13363360000 - 4 \cdot 49 \cdot - 6938890000}}{2 \cdot 49}$

Step 6.1.2

Multiply $- 4 \cdot 49 \cdot - 6938890000$ .

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Step 6.1.2.1

Multiply $- 4$ by $49$ .

$d = \frac{- 115600 \pm \sqrt{13363360000 - 196 \cdot - 6938890000}}{2 \cdot 49}$

Step 6.1.2.2

Multiply $- 196$ by $- 6938890000$ .

$d = \frac{- 115600 \pm \sqrt{13363360000 + 1360022440000}}{2 \cdot 49}$

Step 6.1.3

Add $13363360000$ and $1360022440000$ .

$d = \frac{- 115600 \pm \sqrt{1373385800000}}{2 \cdot 49}$

Step 6.1.4

Rewrite $1373385800000$ as $3400^{2} \cdot 118805$ .

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Step 6.1.4.1

Factor $11560000$ out of $1373385800000$ .

$d = \frac{- 115600 \pm \sqrt{11560000 (118805)}}{2 \cdot 49}$

Step 6.1.4.2

Rewrite $11560000$ as $3400^{2}$ .

$d = \frac{- 115600 \pm \sqrt{3400^{2} \cdot 118805}}{2 \cdot 49}$

Step 6.1.5

Pull terms out from under the radical.

$d = \frac{- 115600 \pm 3400 \sqrt{118805}}{2 \cdot 49}$

Step 6.2

Multiply $2$ by $49$ .

$d = \frac{- 115600 \pm 3400 \sqrt{118805}}{98}$

Step 6.3

Simplify $\frac{- 115600 \pm 3400 \sqrt{118805}}{98}$ .

$d = \frac{- 57800 \pm 1700 \sqrt{118805}}{49}$

Step 7

The final answer is the combination of both solutions.

$d = - \frac{57800 - 1700 \sqrt{118805}}{49}, - \frac{57800 + 1700 \sqrt{118805}}{49}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$d = - \frac{57800 - 1700 \sqrt{118805}}{49}, - \frac{57800 + 1700 \sqrt{118805}}{49}$

Decimal Form:

$d = 10778.72898607 \dots, - 13137.91265954 \dots$