Calculus Examples

$(9 x + 4) (12 x - 15) = (2 x - 3) (5 - 4 x)$

Step 1

Simplify $(9 x + 4) (12 x - 15)$ .

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Step 1.1

Rewrite.

$0 + 0 + (9 x + 4) (12 x - 15) = (2 x - 3) (5 - 4 x)$

Step 1.2

Simplify by adding zeros.

$(9 x + 4) (12 x - 15) = (2 x - 3) (5 - 4 x)$

Step 1.3

Expand $(9 x + 4) (12 x - 15)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$9 x (12 x - 15) + 4 (12 x - 15) = (2 x - 3) (5 - 4 x)$

Step 1.3.2

Apply the distributive property.

$9 x (12 x) + 9 x \cdot - 15 + 4 (12 x - 15) = (2 x - 3) (5 - 4 x)$

Step 1.3.3

Apply the distributive property.

$9 x (12 x) + 9 x \cdot - 15 + 4 (12 x) + 4 \cdot - 15 = (2 x - 3) (5 - 4 x)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$9 \cdot 12 x \cdot x + 9 x \cdot - 15 + 4 (12 x) + 4 \cdot - 15 = (2 x - 3) (5 - 4 x)$

Step 1.4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.2.1

Move $x$ .

$9 \cdot 12 (x \cdot x) + 9 x \cdot - 15 + 4 (12 x) + 4 \cdot - 15 = (2 x - 3) (5 - 4 x)$

Step 1.4.1.2.2

Multiply $x$ by $x$ .

$9 \cdot 12 x^{2} + 9 x \cdot - 15 + 4 (12 x) + 4 \cdot - 15 = (2 x - 3) (5 - 4 x)$

Step 1.4.1.3

Multiply $9$ by $12$ .

$108 x^{2} + 9 x \cdot - 15 + 4 (12 x) + 4 \cdot - 15 = (2 x - 3) (5 - 4 x)$

Step 1.4.1.4

Multiply $- 15$ by $9$ .

$108 x^{2} - 135 x + 4 (12 x) + 4 \cdot - 15 = (2 x - 3) (5 - 4 x)$

Step 1.4.1.5

Multiply $12$ by $4$ .

$108 x^{2} - 135 x + 48 x + 4 \cdot - 15 = (2 x - 3) (5 - 4 x)$

Step 1.4.1.6

Multiply $4$ by $- 15$ .

$108 x^{2} - 135 x + 48 x - 60 = (2 x - 3) (5 - 4 x)$

Step 1.4.2

Add $- 135 x$ and $48 x$ .

$108 x^{2} - 87 x - 60 = (2 x - 3) (5 - 4 x)$

Step 2

Simplify $(2 x - 3) (5 - 4 x)$ .

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Step 2.1

Expand $(2 x - 3) (5 - 4 x)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$108 x^{2} - 87 x - 60 = 2 x (5 - 4 x) - 3 (5 - 4 x)$

Step 2.1.2

Apply the distributive property.

$108 x^{2} - 87 x - 60 = 2 x \cdot 5 + 2 x (- 4 x) - 3 (5 - 4 x)$

Step 2.1.3

Apply the distributive property.

$108 x^{2} - 87 x - 60 = 2 x \cdot 5 + 2 x (- 4 x) - 3 \cdot 5 - 3 (- 4 x)$

Step 2.2

Simplify and combine like terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $5$ by $2$ .

$108 x^{2} - 87 x - 60 = 10 x + 2 x (- 4 x) - 3 \cdot 5 - 3 (- 4 x)$

Step 2.2.1.2

Rewrite using the commutative property of multiplication.

$108 x^{2} - 87 x - 60 = 10 x + 2 \cdot - 4 x \cdot x - 3 \cdot 5 - 3 (- 4 x)$

Step 2.2.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.3.1

Move $x$ .

$108 x^{2} - 87 x - 60 = 10 x + 2 \cdot - 4 (x \cdot x) - 3 \cdot 5 - 3 (- 4 x)$

Step 2.2.1.3.2

Multiply $x$ by $x$ .

$108 x^{2} - 87 x - 60 = 10 x + 2 \cdot - 4 x^{2} - 3 \cdot 5 - 3 (- 4 x)$

Step 2.2.1.4

Multiply $2$ by $- 4$ .

$108 x^{2} - 87 x - 60 = 10 x - 8 x^{2} - 3 \cdot 5 - 3 (- 4 x)$

Step 2.2.1.5

Multiply $- 3$ by $5$ .

$108 x^{2} - 87 x - 60 = 10 x - 8 x^{2} - 15 - 3 (- 4 x)$

Step 2.2.1.6

Multiply $- 4$ by $- 3$ .

$108 x^{2} - 87 x - 60 = 10 x - 8 x^{2} - 15 + 12 x$

Step 2.2.2

Add $10 x$ and $12 x$ .

$108 x^{2} - 87 x - 60 = 22 x - 8 x^{2} - 15$

Step 3

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

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Step 3.1

Subtract $22 x$ from both sides of the equation.

$108 x^{2} - 87 x - 60 - 22 x = - 8 x^{2} - 15$

Step 3.2

Add $8 x^{2}$ to both sides of the equation.

$108 x^{2} - 87 x - 60 - 22 x + 8 x^{2} = - 15$

Step 3.3

Add $108 x^{2}$ and $8 x^{2}$ .

$116 x^{2} - 87 x - 60 - 22 x = - 15$

Step 3.4

Subtract $22 x$ from $- 87 x$ .

$116 x^{2} - 109 x - 60 = - 15$

Step 4

Add $15$ to both sides of the equation.

$116 x^{2} - 109 x - 60 + 15 = 0$

Step 5

Add $- 60$ and $15$ .

$116 x^{2} - 109 x - 45 = 0$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 116 \cdot - 45 = - 5220$ and whose sum is $b = - 109$ .

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Step 6.1.1

Factor $- 109$ out of $- 109 x$ .

$116 x^{2} - 109 x - 45 = 0$

Step 6.1.2

Rewrite $- 109$ as $36$ plus $- 145$

$116 x^{2} + (36 - 145) x - 45 = 0$

Step 6.1.3

Apply the distributive property.

$116 x^{2} + 36 x - 145 x - 45 = 0$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$(116 x^{2} + 36 x) - 145 x - 45 = 0$

Step 6.2.2

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

$4 x (29 x + 9) - 5 (29 x + 9) = 0$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $29 x + 9$ .

$(29 x + 9) (4 x - 5) = 0$

Step 7

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

$29 x + 9 = 0$

$4 x - 5 = 0$

Step 8

Set $29 x + 9$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $29 x + 9$ equal to $0$ .

$29 x + 9 = 0$

Step 8.2

Solve $29 x + 9 = 0$ for $x$ .

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Step 8.2.1

Subtract $9$ from both sides of the equation.

$29 x = - 9$

Step 8.2.2

Divide each term in $29 x = - 9$ by $29$ and simplify.

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Step 8.2.2.1

Divide each term in $29 x = - 9$ by $29$ .

$\frac{29 x}{29} = \frac{- 9}{29}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $29$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{29 x}{29} = \frac{- 9}{29}$

Step 8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{29}$

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.

$x = - \frac{9}{29}$

Step 9

Set $4 x - 5$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $4 x - 5$ equal to $0$ .

$4 x - 5 = 0$

Step 9.2

Solve $4 x - 5 = 0$ for $x$ .

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Step 9.2.1

Add $5$ to both sides of the equation.

$4 x = 5$

Step 9.2.2

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

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Step 9.2.2.1

Divide each term in $4 x = 5$ by $4$ .

$\frac{4 x}{4} = \frac{5}{4}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $4$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{5}{4}$

Step 9.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{4}$

Step 10

The final solution is all the values that make $(29 x + 9) (4 x - 5) = 0$ true.

$x = - \frac{9}{29}, \frac{5}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{9}{29}, \frac{5}{4}$

Decimal Form:

$x = - 0.31034482 \dots, 1.25$

Mixed Number Form:

$x = - \frac{9}{29}, 1 \frac{1}{4}$