Finite Math Examples

$5 u^{2} - 24 u = 5$

Step 1

Subtract $5$ from both sides of the equation.

$5 u^{2} - 24 u - 5 = 0$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 5 \cdot - 5 = - 25$ and whose sum is $b = - 24$ .

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Step 2.1.1

Factor $- 24$ out of $- 24 u$ .

$5 u^{2} - 24 u - 5 = 0$

Step 2.1.2

Rewrite $- 24$ as $1$ plus $- 25$

$5 u^{2} + (1 - 25) u - 5 = 0$

Step 2.1.3

Apply the distributive property.

$5 u^{2} + 1 u - 25 u - 5 = 0$

Step 2.1.4

Multiply $u$ by $1$ .

$5 u^{2} + u - 25 u - 5 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(5 u^{2} + u) - 25 u - 5 = 0$

Step 2.2.2

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

$u (5 u + 1) - 5 (5 u + 1) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $5 u + 1$ .

$(5 u + 1) (u - 5) = 0$

Step 3

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

$5 u + 1 = 0$

$u - 5 = 0$

Step 4

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

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Step 4.1

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

$5 u + 1 = 0$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$5 u = - 1$

Step 4.2.2

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

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Step 4.2.2.1

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

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

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.

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

Step 4.2.2.2.1.2

Divide $u$ by $1$ .

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

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.

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

Step 5

Set $u - 5$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

Step 5.2

Add $5$ to both sides of the equation.

$u = 5$

Step 6

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

$u = - \frac{1}{5}, 5$

Step 7

The result can be shown in multiple forms.

Exact Form:

$u = - \frac{1}{5}, 5$

Decimal Form:

$u = - 0.2, 5$