Algebra Examples

$10 = 3.1 + 40 (\frac{x}{140}) - 16 {(\frac{x}{140})}^{2}$

Step 1

Rewrite the equation as $3.1 + 40 (\frac{x}{140}) - 16 {(\frac{x}{140})}^{2} = 10$ .

$3.1 + 40 (\frac{x}{140}) - 16 {(\frac{x}{140})}^{2} = 10$

Step 2

Simplify each term.

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Step 2.1

Cancel the common factor of $20$ .

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Step 2.1.1

Factor $20$ out of $40$ .

$3.1 + 20 (2) \frac{x}{140} - 16 {(\frac{x}{140})}^{2} = 10$

Step 2.1.2

Factor $20$ out of $140$ .

$3.1 + 20 \cdot 2 \frac{x}{20 \cdot 7} - 16 {(\frac{x}{140})}^{2} = 10$

Step 2.1.3

Cancel the common factor.

$3.1 + 20 \cdot 2 \frac{x}{20 \cdot 7} - 16 {(\frac{x}{140})}^{2} = 10$

Step 2.1.4

Rewrite the expression.

$3.1 + 2 \frac{x}{7} - 16 {(\frac{x}{140})}^{2} = 10$

Step 2.2

Combine $2$ and $\frac{x}{7}$ .

$3.1 + \frac{2 x}{7} - 16 {(\frac{x}{140})}^{2} = 10$

Step 2.3

Apply the product rule to $\frac{x}{140}$ .

$3.1 + \frac{2 x}{7} - 16 \frac{x^{2}}{140^{2}} = 10$

Step 2.4

Raise $140$ to the power of $2$ .

$3.1 + \frac{2 x}{7} - 16 \frac{x^{2}}{19600} = 10$

Step 2.5

Cancel the common factor of $16$ .

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Step 2.5.1

Factor $16$ out of $- 16$ .

$3.1 + \frac{2 x}{7} + 16 (- 1) \frac{x^{2}}{19600} = 10$

Step 2.5.2

Factor $16$ out of $19600$ .

$3.1 + \frac{2 x}{7} + 16 \cdot - 1 \frac{x^{2}}{16 \cdot 1225} = 10$

Step 2.5.3

Cancel the common factor.

$3.1 + \frac{2 x}{7} + 16 \cdot - 1 \frac{x^{2}}{16 \cdot 1225} = 10$

Step 2.5.4

Rewrite the expression.

$3.1 + \frac{2 x}{7} - 1 \frac{x^{2}}{1225} = 10$

Step 2.6

Rewrite $- 1 \frac{x^{2}}{1225}$ as $- \frac{x^{2}}{1225}$ .

$3.1 + \frac{2 x}{7} - \frac{x^{2}}{1225} = 10$

Step 3

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

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Step 3.1

Subtract $10$ from both sides of the equation.

$3.1 + \frac{2 x}{7} - \frac{x^{2}}{1225} - 10 = 0$

Step 3.2

Subtract $10$ from $3.1$ .

$\frac{2 x}{7} - \frac{x^{2}}{1225} - 6.9 = 0$

Step 4

Multiply through by the least common denominator $1225$ , then simplify.

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Step 4.1

Apply the distributive property.

$1225 (\frac{2 x}{7}) + 1225 (- \frac{x^{2}}{1225}) + 1225 \cdot - 6.9 = 0$

Step 4.2

Simplify.

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Step 4.2.1

Cancel the common factor of $7$ .

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Step 4.2.1.1

Factor $7$ out of $1225$ .

$7 (175) (\frac{2 x}{7}) + 1225 (- \frac{x^{2}}{1225}) + 1225 \cdot - 6.9 = 0$

Step 4.2.1.2

Cancel the common factor.

$7 \cdot (175 (\frac{2 x}{7})) + 1225 (- \frac{x^{2}}{1225}) + 1225 \cdot - 6.9 = 0$

Step 4.2.1.3

Rewrite the expression.

$175 (2 x) + 1225 (- \frac{x^{2}}{1225}) + 1225 \cdot - 6.9 = 0$

Step 4.2.2

Multiply $2$ by $175$ .

$350 x + 1225 (- \frac{x^{2}}{1225}) + 1225 \cdot - 6.9 = 0$

Step 4.2.3

Cancel the common factor of $1225$ .

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Step 4.2.3.1

Move the leading negative in $- \frac{x^{2}}{1225}$ into the numerator.

$350 x + 1225 (\frac{- x^{2}}{1225}) + 1225 \cdot - 6.9 = 0$

Step 4.2.3.2

Cancel the common factor.

$350 x + 1225 (\frac{- x^{2}}{1225}) + 1225 \cdot - 6.9 = 0$

Step 4.2.3.3

Rewrite the expression.

$350 x - x^{2} + 1225 \cdot - 6.9 = 0$

Step 4.2.4

Multiply $1225$ by $- 6.9$ .

$350 x - x^{2} - 8452.5 = 0$

Step 4.3

Reorder $350 x$ and $- x^{2}$ .

$- x^{2} + 350 x - 8452.5 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 1$ , $b = 350$ , and $c = - 8452.5$ into the quadratic formula and solve for $x$ .

$\frac{- 350 \pm \sqrt{350^{2} - 4 \cdot (- 1 \cdot - 8452.5)}}{2 \cdot - 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $350$ to the power of $2$ .

$x = \frac{- 350 \pm \sqrt{122500 - 4 \cdot - 1 \cdot - 8452.5}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 8452.5$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 350 \pm \sqrt{122500 + 4 \cdot - 8452.5}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 8452.5$ .

$x = \frac{- 350 \pm \sqrt{122500 - 33810}}{2 \cdot - 1}$

Step 7.1.3

Subtract $33810$ from $122500$ .

$x = \frac{- 350 \pm \sqrt{88690}}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 350 \pm \sqrt{88690}}{- 2}$

Step 7.3

Simplify $\frac{- 350 \pm \sqrt{88690}}{- 2}$ .

$x = \frac{350 \pm \sqrt{88690}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{350 + \sqrt{88690}}{2}, \frac{350 - \sqrt{88690}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{350 + \sqrt{88690}}{2}, \frac{350 - \sqrt{88690}}{2}$

Decimal Form:

$x = 323.90433170 \dots, 26.09566829 \dots$