Finite Math Examples

$24 c^{2} (2 c - 7) = - 35 (2 c - 7) - 58 c (2 c - 7)$

Step 1

Since $c$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 35 (2 c - 7) - 58 c (2 c - 7) = 24 c^{2} (2 c - 7)$

Step 2

Simplify $- 35 (2 c - 7) - 58 c (2 c - 7)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- 35 (2 c) - 35 \cdot - 7 - 58 c (2 c - 7) = 24 c^{2} (2 c - 7)$

Step 2.1.2

Multiply $2$ by $- 35$ .

$- 70 c - 35 \cdot - 7 - 58 c (2 c - 7) = 24 c^{2} (2 c - 7)$

Step 2.1.3

Multiply $- 35$ by $- 7$ .

$- 70 c + 245 - 58 c (2 c - 7) = 24 c^{2} (2 c - 7)$

Step 2.1.4

Apply the distributive property.

$- 70 c + 245 - 58 c (2 c) - 58 c \cdot - 7 = 24 c^{2} (2 c - 7)$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$- 70 c + 245 - 58 \cdot 2 c \cdot c - 58 c \cdot - 7 = 24 c^{2} (2 c - 7)$

Step 2.1.6

Multiply $- 7$ by $- 58$ .

$- 70 c + 245 - 58 \cdot 2 c \cdot c + 406 c = 24 c^{2} (2 c - 7)$

Step 2.1.7

Simplify each term.

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Step 2.1.7.1

Multiply $c$ by $c$ by adding the exponents.

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Step 2.1.7.1.1

Move $c$ .

$- 70 c + 245 - 58 \cdot 2 (c \cdot c) + 406 c = 24 c^{2} (2 c - 7)$

Step 2.1.7.1.2

Multiply $c$ by $c$ .

$- 70 c + 245 - 58 \cdot 2 c^{2} + 406 c = 24 c^{2} (2 c - 7)$

Step 2.1.7.2

Multiply $- 58$ by $2$ .

$- 70 c + 245 - 116 c^{2} + 406 c = 24 c^{2} (2 c - 7)$

Step 2.2

Add $- 70 c$ and $406 c$ .

$336 c + 245 - 116 c^{2} = 24 c^{2} (2 c - 7)$

Step 3

Simplify $24 c^{2} (2 c - 7)$ .

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Step 3.1

Simplify by multiplying through.

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Step 3.1.1

Apply the distributive property.

$336 c + 245 - 116 c^{2} = 24 c^{2} (2 c) + 24 c^{2} \cdot - 7$

Step 3.1.2

Simplify the expression.

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Step 3.1.2.1

Rewrite using the commutative property of multiplication.

$336 c + 245 - 116 c^{2} = 24 \cdot 2 c^{2} c + 24 c^{2} \cdot - 7$

Step 3.1.2.2

Multiply $- 7$ by $24$ .

$336 c + 245 - 116 c^{2} = 24 \cdot 2 c^{2} c - 168 c^{2}$

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $c^{2}$ by $c$ by adding the exponents.

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Step 3.2.1.1

Move $c$ .

$336 c + 245 - 116 c^{2} = 24 \cdot 2 (c \cdot c^{2}) - 168 c^{2}$

Step 3.2.1.2

Multiply $c$ by $c^{2}$ .

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Step 3.2.1.2.1

Raise $c$ to the power of $1$ .

$336 c + 245 - 116 c^{2} = 24 \cdot 2 (c^{1} c^{2}) - 168 c^{2}$

Step 3.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$336 c + 245 - 116 c^{2} = 24 \cdot 2 c^{1 + 2} - 168 c^{2}$

Step 3.2.1.3

Add $1$ and $2$ .

$336 c + 245 - 116 c^{2} = 24 \cdot 2 c^{3} - 168 c^{2}$

Step 3.2.2

Multiply $24$ by $2$ .

$336 c + 245 - 116 c^{2} = 48 c^{3} - 168 c^{2}$

Step 4

Move all terms containing $c$ to the left side of the equation.

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Step 4.1

Subtract $48 c^{3}$ from both sides of the equation.

$336 c + 245 - 116 c^{2} - 48 c^{3} = - 168 c^{2}$

Step 4.2

Add $168 c^{2}$ to both sides of the equation.

$336 c + 245 - 116 c^{2} - 48 c^{3} + 168 c^{2} = 0$

Step 4.3

Add $- 116 c^{2}$ and $168 c^{2}$ .

$336 c + 245 - 48 c^{3} + 52 c^{2} = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Reorder terms.

$- 48 c^{3} + 52 c^{2} + 336 c + 245 = 0$

Step 5.2

Factor $- 48 c^{3} + 52 c^{2} + 336 c + 245$ using the rational roots test.

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Step 5.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 245, \pm 5, \pm 49, \pm 7, \pm 35$

$q = \pm 1, \pm 48, \pm 2, \pm 24, \pm 3, \pm 16, \pm 4, \pm 12, \pm 6, \pm 8$

Step 5.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.0208\overline{3}, \pm 0.5, \pm 0.041\overline{6}, \pm 0.\overline{3}, \pm 0.0625, \pm 0.25, \pm 0.08\overline{3}, \pm 0.1\overline{6}, \pm 0.125, \pm 245, \pm 5.1041\overline{6}, \pm 122.5, \pm 10.208\overline{3}, \pm 81.\overline{6}, \pm 15.3125, \pm 61.25, \pm 20.41\overline{6}, \pm 40.8\overline{3}, \pm 30.625, \pm 5, \pm 0.1041\overline{6}, \pm 2.5, \pm 0.208\overline{3}, \pm 1.\overline{6}, \pm 0.3125, \pm 1.25, \pm 0.41\overline{6}, \pm 0.8\overline{3}, \pm 0.625, \pm 49, \pm 1.0208\overline{3}, \pm 24.5, \pm 2.041\overline{6}, \pm 16.\overline{3}, \pm 3.0625, \pm 12.25, \pm 4.08\overline{3}, \pm 8.1\overline{6}, \pm 6.125, \pm 7, \pm 0.1458\overline{3}, \pm 3.5, \pm 0.291\overline{6}, \pm 2.\overline{3}, \pm 0.4375, \pm 1.75, \pm 0.58\overline{3}, \pm 1.1\overline{6}, \pm 0.875, \pm 35, \pm 0.7291\overline{6}, \pm 17.5, \pm 1.458\overline{3}, \pm 11.\overline{6}, \pm 2.1875, \pm 8.75, \pm 2.91\overline{6}, \pm 5.8\overline{3}, \pm 4.375$

Step 5.2.3

Substitute $- 1.25$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1.25$ is a root of the polynomial.

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Step 5.2.3.1

Substitute $- 1.25$ into the polynomial.

$- 48 {(- 1.25)}^{3} + 52 {(- 1.25)}^{2} + 336 \cdot - 1.25 + 245$

Step 5.2.3.2

Raise $- 1.25$ to the power of $3$ .

$- 48 \cdot - 1.953125 + 52 {(- 1.25)}^{2} + 336 \cdot - 1.25 + 245$

Step 5.2.3.3

Multiply $- 48$ by $- 1.953125$ .

$93.75 + 52 {(- 1.25)}^{2} + 336 \cdot - 1.25 + 245$

Step 5.2.3.4

Raise $- 1.25$ to the power of $2$ .

$93.75 + 52 \cdot 1.5625 + 336 \cdot - 1.25 + 245$

Step 5.2.3.5

Multiply $52$ by $1.5625$ .

$93.75 + 81.25 + 336 \cdot - 1.25 + 245$

Step 5.2.3.6

Add $93.75$ and $81.25$ .

$175 + 336 \cdot - 1.25 + 245$

Step 5.2.3.7

Multiply $336$ by $- 1.25$ .

$175 - 420 + 245$

Step 5.2.3.8

Subtract $420$ from $175$ .

$- 245 + 245$

Step 5.2.3.9

Add $- 245$ and $245$ .

$0$

Step 5.2.4

Since $- 1.25$ is a known root, divide the polynomial by $4 c + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- 48 c^{3} + 52 c^{2} + 336 c + 245}{4 c + 5}$

Step 5.2.5

Divide $- 48 c^{3} + 52 c^{2} + 336 c + 245$ by $4 c + 5$ .

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Step 5.2.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$4 c$

$5$

$48 c^{3}$

$52 c^{2}$

$336 c$

$245$

Step 5.2.5.2

Divide the highest order term in the dividend $- 48 c^{3}$ by the highest order term in divisor $4 c$ .

				-	$12 c^{2}$
	$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$

Step 5.2.5.3

Multiply the new quotient term by the divisor.

			-	$12 c^{2}$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			-	$48 c^{3}$	-	$60 c^{2}$

Step 5.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- 48 c^{3} - 60 c^{2}$

			-	$12 c^{2}$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$

Step 5.2.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$12 c^{2}$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$

Step 5.2.5.6

Pull the next terms from the original dividend down into the current dividend.

			-	$12 c^{2}$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$

Step 5.2.5.7

Divide the highest order term in the dividend $112 c^{2}$ by the highest order term in divisor $4 c$ .

			-	$12 c^{2}$	+	$28 c$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$

Step 5.2.5.8

Multiply the new quotient term by the divisor.

			-	$12 c^{2}$	+	$28 c$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					+	$112 c^{2}$	+	$140 c$

Step 5.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $112 c^{2} + 140 c$

			-	$12 c^{2}$	+	$28 c$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					-	$112 c^{2}$	-	$140 c$

Step 5.2.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$12 c^{2}$	+	$28 c$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					-	$112 c^{2}$	-	$140 c$
							+	$196 c$

Step 5.2.5.11

Pull the next terms from the original dividend down into the current dividend.

			-	$12 c^{2}$	+	$28 c$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					-	$112 c^{2}$	-	$140 c$
							+	$196 c$	+	$245$

Step 5.2.5.12

Divide the highest order term in the dividend $196 c$ by the highest order term in divisor $4 c$ .

			-	$12 c^{2}$	+	$28 c$	+	$49$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					-	$112 c^{2}$	-	$140 c$
							+	$196 c$	+	$245$

Step 5.2.5.13

Multiply the new quotient term by the divisor.

			-	$12 c^{2}$	+	$28 c$	+	$49$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					-	$112 c^{2}$	-	$140 c$
							+	$196 c$	+	$245$
							+	$196 c$	+	$245$

Step 5.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $196 c + 245$

			-	$12 c^{2}$	+	$28 c$	+	$49$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					-	$112 c^{2}$	-	$140 c$
							+	$196 c$	+	$245$
							-	$196 c$	-	$245$

Step 5.2.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$12 c^{2}$	+	$28 c$	+	$49$
$4 c$	+	$5$	-	$48 c^{3}$	+	$52 c^{2}$	+	$336 c$	+	$245$
			+	$48 c^{3}$	+	$60 c^{2}$
					+	$112 c^{2}$	+	$336 c$
					-	$112 c^{2}$	-	$140 c$
							+	$196 c$	+	$245$
							-	$196 c$	-	$245$
										$0$

Step 5.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$- 12 c^{2} + 28 c + 49$

Step 5.2.6

Write $- 48 c^{3} + 52 c^{2} + 336 c + 245$ as a set of factors.

$(4 c + 5) (- 12 c^{2} + 28 c + 49) = 0$

Step 5.3

Factor.

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Step 5.3.1

Factor by grouping.

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Step 5.3.1.1

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 $a \cdot c = - 12 \cdot 49 = - 588$ and whose sum is $b = 28$ .

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Step 5.3.1.1.1

Factor $28$ out of $28 c$ .

$(4 c + 5) (- 12 c^{2} + 28 (c) + 49) = 0$

Step 5.3.1.1.2

Rewrite $28$ as $- 14$ plus $42$

$(4 c + 5) (- 12 c^{2} + (- 14 + 42) c + 49) = 0$

Step 5.3.1.1.3

Apply the distributive property.

$(4 c + 5) (- 12 c^{2} - 14 c + 42 c + 49) = 0$

Step 5.3.1.2

Factor out the greatest common factor from each group.

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Step 5.3.1.2.1

Group the first two terms and the last two terms.

$(4 c + 5) ((- 12 c^{2} - 14 c) + 42 c + 49) = 0$

Step 5.3.1.2.2

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

$(4 c + 5) (2 c (- 6 c - 7) - 7 (- 6 c - 7)) = 0$

Step 5.3.1.3

Factor the polynomial by factoring out the greatest common factor, $- 6 c - 7$ .

$(4 c + 5) ((- 6 c - 7) (2 c - 7)) = 0$

Step 5.3.2

Remove unnecessary parentheses.

$(4 c + 5) (- 6 c - 7) (2 c - 7) = 0$

Step 6

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

$4 c + 5 = 0$

$- 6 c - 7 = 0$

$2 c - 7 = 0$

Step 7

Set $4 c + 5$ equal to $0$ and solve for $c$ .

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Step 7.1

Set $4 c + 5$ equal to $0$ .

$4 c + 5 = 0$

Step 7.2

Solve $4 c + 5 = 0$ for $c$ .

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Step 7.2.1

Subtract $5$ from both sides of the equation.

$4 c = - 5$

Step 7.2.2

Divide each term in $4 c = - 5$ by $4$ and simplify.

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Step 7.2.2.1

Divide each term in $4 c = - 5$ by $4$ .

$\frac{4 c}{4} = \frac{- 5}{4}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $4$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{4 c}{4} = \frac{- 5}{4}$

Step 7.2.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{- 5}{4}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$c = - \frac{5}{4}$

Step 8

Set $- 6 c - 7$ equal to $0$ and solve for $c$ .

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Step 8.1

Set $- 6 c - 7$ equal to $0$ .

$- 6 c - 7 = 0$

Step 8.2

Solve $- 6 c - 7 = 0$ for $c$ .

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Step 8.2.1

Add $7$ to both sides of the equation.

$- 6 c = 7$

Step 8.2.2

Divide each term in $- 6 c = 7$ by $- 6$ and simplify.

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Step 8.2.2.1

Divide each term in $- 6 c = 7$ by $- 6$ .

$\frac{- 6 c}{- 6} = \frac{7}{- 6}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $- 6$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{- 6 c}{- 6} = \frac{7}{- 6}$

Step 8.2.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{7}{- 6}$

Step 8.2.2.3

Simplify the right side.

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Step 8.2.2.3.1

Move the negative in front of the fraction.

$c = - \frac{7}{6}$

Step 9

Set $2 c - 7$ equal to $0$ and solve for $c$ .

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Step 9.1

Set $2 c - 7$ equal to $0$ .

$2 c - 7 = 0$

Step 9.2

Solve $2 c - 7 = 0$ for $c$ .

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Step 9.2.1

Add $7$ to both sides of the equation.

$2 c = 7$

Step 9.2.2

Divide each term in $2 c = 7$ by $2$ and simplify.

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Step 9.2.2.1

Divide each term in $2 c = 7$ by $2$ .

$\frac{2 c}{2} = \frac{7}{2}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $2$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{2 c}{2} = \frac{7}{2}$

Step 9.2.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{7}{2}$

Step 10

The final solution is all the values that make $(4 c + 5) (- 6 c - 7) (2 c - 7) = 0$ true.

$c = - \frac{5}{4}, - \frac{7}{6}, \frac{7}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$c = - \frac{5}{4}, - \frac{7}{6}, \frac{7}{2}$

Decimal Form:

$c = - 1.25, - 1.1\overline{6}, 3.5$

Mixed Number Form:

$c = - 1 \frac{1}{4}, - 1 \frac{1}{6}, 3 \frac{1}{2}$