Basic Math Examples

$75 d^{3} + 50 d^{2} - 3 d - 2 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$(75 d^{3} + 50 d^{2}) - 3 d - 2 = 0$

Step 1.1.2

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

$25 d^{2} (3 d + 2) - (3 d + 2) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $3 d + 2$ .

$(3 d + 2) (25 d^{2} - 1) = 0$

Step 1.3

Rewrite $25 d^{2}$ as ${(5 d)}^{2}$ .

$(3 d + 2) ({(5 d)}^{2} - 1) = 0$

Step 1.4

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

$(3 d + 2) ({(5 d)}^{2} - 1^{2}) = 0$

Step 1.5

Factor.

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Step 1.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 d$ and $b = 1$ .

$(3 d + 2) ((5 d + 1) (5 d - 1)) = 0$

Step 1.5.2

Remove unnecessary parentheses.

$(3 d + 2) (5 d + 1) (5 d - 1) = 0$

Step 2

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

$3 d + 2 = 0$

$5 d + 1 = 0$

$5 d - 1 = 0$

Step 3

Set $3 d + 2$ equal to $0$ and solve for $d$ .

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Step 3.1

Set $3 d + 2$ equal to $0$ .

$3 d + 2 = 0$

Step 3.2

Solve $3 d + 2 = 0$ for $d$ .

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Step 3.2.1

Subtract $2$ from both sides of the equation.

$3 d = - 2$

Step 3.2.2

Divide each term in $3 d = - 2$ by $3$ and simplify.

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Step 3.2.2.1

Divide each term in $3 d = - 2$ by $3$ .

$\frac{3 d}{3} = \frac{- 2}{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 d}{3} = \frac{- 2}{3}$

Step 3.2.2.2.1.2

Divide $d$ by $1$ .

$d = \frac{- 2}{3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

$d = - \frac{2}{3}$

Step 4

Set $5 d + 1$ equal to $0$ and solve for $d$ .

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Step 4.1

Set $5 d + 1$ equal to $0$ .

$5 d + 1 = 0$

Step 4.2

Solve $5 d + 1 = 0$ for $d$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$5 d = - 1$

Step 4.2.2

Divide each term in $5 d = - 1$ by $5$ and simplify.

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Step 4.2.2.1

Divide each term in $5 d = - 1$ by $5$ .

$\frac{5 d}{5} = \frac{- 1}{5}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{5 d}{5} = \frac{- 1}{5}$

Step 4.2.2.2.1.2

Divide $d$ by $1$ .

$d = \frac{- 1}{5}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$d = - \frac{1}{5}$

Step 5

Set $5 d - 1$ equal to $0$ and solve for $d$ .

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Step 5.1

Set $5 d - 1$ equal to $0$ .

$5 d - 1 = 0$

Step 5.2

Solve $5 d - 1 = 0$ for $d$ .

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Step 5.2.1

Add $1$ to both sides of the equation.

$5 d = 1$

Step 5.2.2

Divide each term in $5 d = 1$ by $5$ and simplify.

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Step 5.2.2.1

Divide each term in $5 d = 1$ by $5$ .

$\frac{5 d}{5} = \frac{1}{5}$

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.

$\frac{5 d}{5} = \frac{1}{5}$

Step 5.2.2.2.1.2

Divide $d$ by $1$ .

$d = \frac{1}{5}$

Step 6

The final solution is all the values that make $(3 d + 2) (5 d + 1) (5 d - 1) = 0$ true.

$d = - \frac{2}{3}, - \frac{1}{5}, \frac{1}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$d = - \frac{2}{3}, - \frac{1}{5}, \frac{1}{5}$

Decimal Form:

$d = - 0.\overline{6}, - 0.2, 0.2$