Algebra Examples

$\frac{x^{2}}{2} + \frac{13 x}{20} = \frac{3}{2}$

Step 1

Multiply each term in

$\frac{x^{2}}{2} + \frac{13 x}{20} = \frac{3}{2}$ by

$20$ to eliminate the fractions.

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Step 1.1

Multiply each term in

$\frac{x^{2}}{2} + \frac{13 x}{20} = \frac{3}{2}$ by

$20$ .

$\frac{x^{2}}{2} \cdot 20 + \frac{13 x}{20} \cdot 20 = \frac{3}{2} \cdot 20$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Cancel the common factor of

$2$ .

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Step 1.2.1.1.1

Factor

$2$ out of

$20$ .

$\frac{x^{2}}{2} \cdot (2 (10)) + \frac{13 x}{20} \cdot 20 = \frac{3}{2} \cdot 20$

Step 1.2.1.1.2

Cancel the common factor.

$\frac{x^{2}}{2} \cdot (2 \cdot 10) + \frac{13 x}{20} \cdot 20 = \frac{3}{2} \cdot 20$

Step 1.2.1.1.3

Rewrite the expression.

$x^{2} \cdot 10 + \frac{13 x}{20} \cdot 20 = \frac{3}{2} \cdot 20$

Step 1.2.1.2

Move

$10$ to the left of

$x^{2}$ .

$10 \cdot x^{2} + \frac{13 x}{20} \cdot 20 = \frac{3}{2} \cdot 20$

Step 1.2.1.3

Cancel the common factor of

$20$ .

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Step 1.2.1.3.1

Cancel the common factor.

$10 x^{2} + \frac{13 x}{20} \cdot 20 = \frac{3}{2} \cdot 20$

Step 1.2.1.3.2

Rewrite the expression.

$10 x^{2} + 13 x = \frac{3}{2} \cdot 20$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of

$2$ .

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Step 1.3.1.1

Factor

$2$ out of

$20$ .

$10 x^{2} + 13 x = \frac{3}{2} \cdot (2 (10))$

Step 1.3.1.2

Cancel the common factor.

$10 x^{2} + 13 x = \frac{3}{2} \cdot (2 \cdot 10)$

Step 1.3.1.3

Rewrite the expression.

$10 x^{2} + 13 x = 3 \cdot 10$

Step 1.3.2

Multiply

$3$ by

$10$ .

$10 x^{2} + 13 x = 30$

Step 2

Subtract

$30$ from both sides of the equation.

$10 x^{2} + 13 x - 30 = 0$

Step 3

Factor by grouping.

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Step 3.1

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

$a \cdot c = 10 \cdot - 30 = - 300$ and whose sum is

$b = 13$ .

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Step 3.1.1

Factor

$13$ out of

$13 x$ .

$10 x^{2} + 13 (x) - 30 = 0$

Step 3.1.2

Rewrite

$13$ as

$- 12$ plus

$25$

$10 x^{2} + (- 12 + 25) x - 30 = 0$

Step 3.1.3

Apply the distributive property.

$10 x^{2} - 12 x + 25 x - 30 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(10 x^{2} - 12 x) + 25 x - 30 = 0$

Step 3.2.2

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

$2 x (5 x - 6) + 5 (5 x - 6) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor,

$5 x - 6$ .

$(5 x - 6) (2 x + 5) = 0$

Step 4

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$5 x - 6 = 0$

$2 x + 5 = 0$

Step 5

Set

$5 x - 6$ equal to

$0$ and solve for

$x$ .

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Step 5.1

Set

$5 x - 6$ equal to

$0$ .

$5 x - 6 = 0$

Step 5.2

Solve

$5 x - 6 = 0$ for

$x$ .

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Step 5.2.1

Add

$6$ to both sides of the equation.

$5 x = 6$

Step 5.2.2

Divide each term in

$5 x = 6$ by

$5$ and simplify.

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Step 5.2.2.1

Divide each term in

$5 x = 6$ by

$5$ .

$\frac{5 x}{5} = \frac{6}{5}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of

$5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{6}{5}$

Step 5.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{6}{5}$

Step 6

Set

$2 x + 5$ equal to

$0$ and solve for

$x$ .

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Step 6.1

Set

$2 x + 5$ equal to

$0$ .

$2 x + 5 = 0$

Step 6.2

Solve

$2 x + 5 = 0$ for

$x$ .

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Step 6.2.1

Subtract

$5$ from both sides of the equation.

$2 x = - 5$

Step 6.2.2

Divide each term in

$2 x = - 5$ by

$2$ and simplify.

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Step 6.2.2.1

Divide each term in

$2 x = - 5$ by

$2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of

$2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Step 6.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- 5}{2}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Step 7

The final solution is all the values that make

$(5 x - 6) (2 x + 5) = 0$ true.

$x = \frac{6}{5}, - \frac{5}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{6}{5}, - \frac{5}{2}$

Decimal Form:

$x = 1.2, - 2.5$

Mixed Number Form:

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