Calculus Examples

$a {(x + b)}^{2} = 3 x^{2} + 36 x + c$

Step 1

Rewrite the equation as $3 x^{2} + 36 x + c = a {(x + b)}^{2}$ .

$3 x^{2} + 36 x + c = a {(x + b)}^{2}$

Step 2

Simplify $a {(x + b)}^{2}$ .

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Step 2.1

Rewrite.

$3 x^{2} + 36 x + c = 0 + 0 + a {(x + b)}^{2}$

Step 2.2

Rewrite ${(x + b)}^{2}$ as $(x + b) (x + b)$ .

$3 x^{2} + 36 x + c = a ((x + b) (x + b))$

Step 2.3

Expand $(x + b) (x + b)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$3 x^{2} + 36 x + c = a (x (x + b) + b (x + b))$

Step 2.3.2

Apply the distributive property.

$3 x^{2} + 36 x + c = a (x \cdot x + x b + b (x + b))$

Step 2.3.3

Apply the distributive property.

$3 x^{2} + 36 x + c = a (x \cdot x + x b + b x + b \cdot b)$

Step 2.4

Simplify and combine like terms.

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Step 2.4.1

Simplify each term.

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Step 2.4.1.1

Multiply $x$ by $x$ .

$3 x^{2} + 36 x + c = a (x^{2} + x b + b x + b \cdot b)$

Step 2.4.1.2

Multiply $b$ by $b$ .

$3 x^{2} + 36 x + c = a (x^{2} + x b + b x + b^{2})$

Step 2.4.2

Add $x b$ and $b x$ .

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Step 2.4.2.1

Reorder $x$ and $b$ .

$3 x^{2} + 36 x + c = a (x^{2} + b x + b x + b^{2})$

Step 2.4.2.2

Add $b x$ and $b x$ .

$3 x^{2} + 36 x + c = a (x^{2} + 2 b x + b^{2})$

Step 2.5

Apply the distributive property.

$3 x^{2} + 36 x + c = a x^{2} + a (2 b x) + a b^{2}$

Step 2.6

Rewrite using the commutative property of multiplication.

$3 x^{2} + 36 x + c = a x^{2} + 2 a b x + a b^{2}$

Step 3

Move all terms not containing $c$ to the right side of the equation.

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Step 3.1

Subtract $3 x^{2}$ from both sides of the equation.

$36 x + c = a x^{2} + 2 a b x + a b^{2} - 3 x^{2}$

Step 3.2

Subtract $36 x$ from both sides of the equation.

$c = a x^{2} + 2 a b x + a b^{2} - 3 x^{2} - 36 x$