Finite Math Examples

$- \frac{1}{2} \cdot (32 {(x)}^{2}) + 40 (x) + 80 = 104.64$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $2$ .

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Step 1.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{- 1}{2} \cdot (32 x^{2}) + 40 x + 80 = 104.64$

Step 1.1.2

Factor $2$ out of $32 x^{2}$ .

$\frac{- 1}{2} \cdot (2 (16 x^{2})) + 40 x + 80 = 104.64$

Step 1.1.3

Cancel the common factor.

$\frac{- 1}{2} \cdot (2 (16 x^{2})) + 40 x + 80 = 104.64$

Step 1.1.4

Rewrite the expression.

$- 1 \cdot (16 x^{2}) + 40 x + 80 = 104.64$

Step 1.2

Multiply $16$ by $- 1$ .

$- 16 x^{2} + 40 x + 80 = 104.64$

Step 2

Subtract $104.64$ from both sides of the equation.

$- 16 x^{2} + 40 x + 80 - 104.64 = 0$

Step 3

Subtract $104.64$ from $80$ .

$- 16 x^{2} + 40 x - 24.64 = 0$

Step 4

Factor the left side of the equation.

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Step 4.1

Factor $- 0.32$ out of $- 16 x^{2} + 40 x - 24.64$ .

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Step 4.1.1

Factor $- 0.32$ out of $- 16 x^{2}$ .

$- 0.32 (50 x^{2}) + 40 x - 24.64 = 0$

Step 4.1.2

Factor $- 0.32$ out of $40 x$ .

$- 0.32 (50 x^{2}) - 0.32 (- 125 x) - 24.64 = 0$

Step 4.1.3

Factor $- 0.32$ out of $- 24.64$ .

$- 0.32 (50 x^{2}) - 0.32 (- 125 x) - 0.32 \cdot 77 = 0$

Step 4.1.4

Factor $- 0.32$ out of $- 0.32 (50 x^{2}) - 0.32 (- 125 x)$ .

$- 0.32 (50 x^{2} - 125 x) - 0.32 \cdot 77 = 0$

Step 4.1.5

Factor $- 0.32$ out of $- 0.32 (50 x^{2} - 125 x) - 0.32 (77)$ .

$- 0.32 (50 x^{2} - 125 x + 77) = 0$

Step 4.2

Factor.

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Step 4.2.1

Factor by grouping.

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Step 4.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 50 \cdot 77 = 3850$ and whose sum is $b = - 125$ .

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Step 4.2.1.1.1

Factor $- 125$ out of $- 125 x$ .

$- 0.32 (50 x^{2} - 125 x + 77) = 0$

Step 4.2.1.1.2

Rewrite $- 125$ as $- 55$ plus $- 70$

$- 0.32 (50 x^{2} + (- 55 - 70) x + 77) = 0$

Step 4.2.1.1.3

Apply the distributive property.

$- 0.32 (50 x^{2} - 55 x - 70 x + 77) = 0$

Step 4.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.1.2.1

Group the first two terms and the last two terms.

$- 0.32 ((50 x^{2} - 55 x) - 70 x + 77) = 0$

Step 4.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- 0.32 (5 x (10 x - 11) - 7 (10 x - 11)) = 0$

Step 4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $10 x - 11$ .

$- 0.32 ((10 x - 11) (5 x - 7)) = 0$

Step 4.2.2

Remove unnecessary parentheses.

$- 0.32 (10 x - 11) (5 x - 7) = 0$

Step 5

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

$10 x - 11 = 0$

$5 x - 7 = 0$

Step 6

Set $10 x - 11$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $10 x - 11$ equal to $0$ .

$10 x - 11 = 0$

Step 6.2

Solve $10 x - 11 = 0$ for $x$ .

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Step 6.2.1

Add $11$ to both sides of the equation.

$10 x = 11$

Step 6.2.2

Divide each term in $10 x = 11$ by $10$ and simplify.

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Step 6.2.2.1

Divide each term in $10 x = 11$ by $10$ .

$\frac{10 x}{10} = \frac{11}{10}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $10$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{11}{10}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{10}$

Step 7

Set $5 x - 7$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $5 x - 7$ equal to $0$ .

$5 x - 7 = 0$

Step 7.2

Solve $5 x - 7 = 0$ for $x$ .

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Step 7.2.1

Add $7$ to both sides of the equation.

$5 x = 7$

Step 7.2.2

Divide each term in $5 x = 7$ by $5$ and simplify.

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Step 7.2.2.1

Divide each term in $5 x = 7$ by $5$ .

$\frac{5 x}{5} = \frac{7}{5}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $5$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{7}{5}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{5}$

Step 8

The final solution is all the values that make $- 0.32 (10 x - 11) (5 x - 7) = 0$ true.

$x = \frac{11}{10}, \frac{7}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{10}, \frac{7}{5}$

Decimal Form:

$x = 1.1, 1.4$

Mixed Number Form:

$x = 1 \frac{1}{10}, 1 \frac{2}{5}$