Finite Math Examples

$4 r^{2} - 21 = 25 r$

Step 1

Subtract $25 r$ from both sides of the equation.

$4 r^{2} - 21 - 25 r = 0$

Step 2

Factor by grouping.

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Step 2.1

Reorder terms.

$4 r^{2} - 25 r - 21 = 0$

Step 2.2

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 $a \cdot c = 4 \cdot - 21 = - 84$ and whose sum is $b = - 25$ .

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Step 2.2.1

Factor $- 25$ out of $- 25 r$ .

$4 r^{2} - 25 r - 21 = 0$

Step 2.2.2

Rewrite $- 25$ as $3$ plus $- 28$

$4 r^{2} + (3 - 28) r - 21 = 0$

Step 2.2.3

Apply the distributive property.

$4 r^{2} + 3 r - 28 r - 21 = 0$

Step 2.3

Factor out the greatest common factor from each group.

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Step 2.3.1

Group the first two terms and the last two terms.

$(4 r^{2} + 3 r) - 28 r - 21 = 0$

Step 2.3.2

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

$r (4 r + 3) - 7 (4 r + 3) = 0$

Step 2.4

Factor the polynomial by factoring out the greatest common factor, $4 r + 3$ .

$(4 r + 3) (r - 7) = 0$

Step 3

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

$4 r + 3 = 0$

$r - 7 = 0$

Step 4

Set $4 r + 3$ equal to $0$ and solve for $r$ .

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Step 4.1

Set $4 r + 3$ equal to $0$ .

$4 r + 3 = 0$

Step 4.2

Solve $4 r + 3 = 0$ for $r$ .

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Step 4.2.1

Subtract $3$ from both sides of the equation.

$4 r = - 3$

Step 4.2.2

Divide each term in $4 r = - 3$ by $4$ and simplify.

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Step 4.2.2.1

Divide each term in $4 r = - 3$ by $4$ .

$\frac{4 r}{4} = \frac{- 3}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 r}{4} = \frac{- 3}{4}$

Step 4.2.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 3}{4}$

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.

$r = - \frac{3}{4}$

Step 5

Set $r - 7$ equal to $0$ and solve for $r$ .

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Step 5.1

Set $r - 7$ equal to $0$ .

$r - 7 = 0$

Step 5.2

Add $7$ to both sides of the equation.

$r = 7$

Step 6

The final solution is all the values that make $(4 r + 3) (r - 7) = 0$ true.

$r = - \frac{3}{4}, 7$