Algebra Examples

$4 {(x - 2)}^{2} - 7 = a x^{2} + b x + c$

Step 1

Rewrite the equation as $a x^{2} + b x + c = 4 {(x - 2)}^{2} - 7$ .

$a x^{2} + b x + c = 4 {(x - 2)}^{2} - 7$

Step 2

Simplify $4 {(x - 2)}^{2} - 7$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$a x^{2} + b x + c = 4 ((x - 2) (x - 2)) - 7$

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

$a x^{2} + b x + c = 4 (x (x - 2) - 2 (x - 2)) - 7$

Step 2.1.2.2

Apply the distributive property.

$a x^{2} + b x + c = 4 (x \cdot x + x \cdot - 2 - 2 (x - 2)) - 7$

Step 2.1.2.3

Apply the distributive property.

$a x^{2} + b x + c = 4 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) - 7$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$a x^{2} + b x + c = 4 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) - 7$

Step 2.1.3.1.2

Move $- 2$ to the left of $x$ .

$a x^{2} + b x + c = 4 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) - 7$

Step 2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$a x^{2} + b x + c = 4 (x^{2} - 2 x - 2 x + 4) - 7$

Step 2.1.3.2

Subtract $2 x$ from $- 2 x$ .

$a x^{2} + b x + c = 4 (x^{2} - 4 x + 4) - 7$

Step 2.1.4

Apply the distributive property.

$a x^{2} + b x + c = 4 x^{2} + 4 (- 4 x) + 4 \cdot 4 - 7$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Multiply $- 4$ by $4$ .

$a x^{2} + b x + c = 4 x^{2} - 16 x + 4 \cdot 4 - 7$

Step 2.1.5.2

Multiply $4$ by $4$ .

$a x^{2} + b x + c = 4 x^{2} - 16 x + 16 - 7$

Step 2.2

Subtract $7$ from $16$ .

$a x^{2} + b x + c = 4 x^{2} - 16 x + 9$

Step 3

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

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Step 3.1

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

$a x^{2} + c = 4 x^{2} - 16 x + 9 - b x$

Step 3.2

Subtract $c$ from both sides of the equation.

$a x^{2} = 4 x^{2} - 16 x + 9 - b x - c$

Step 4

Divide each term in $a x^{2} = 4 x^{2} - 16 x + 9 - b x - c$ by $x^{2}$ and simplify.

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Step 4.1

Divide each term in $a x^{2} = 4 x^{2} - 16 x + 9 - b x - c$ by $x^{2}$ .

$\frac{a x^{2}}{x^{2}} = \frac{4 x^{2}}{x^{2}} + \frac{- 16 x}{x^{2}} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $x^{2}$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{a x^{2}}{x^{2}} = \frac{4 x^{2}}{x^{2}} + \frac{- 16 x}{x^{2}} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.2.1.2

Divide $a$ by $1$ .

$a = \frac{4 x^{2}}{x^{2}} + \frac{- 16 x}{x^{2}} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of $x^{2}$ .

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Step 4.3.1.1.1

Cancel the common factor.

$a = \frac{4 x^{2}}{x^{2}} + \frac{- 16 x}{x^{2}} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.1.2

Divide $4$ by $1$ .

$a = 4 + \frac{- 16 x}{x^{2}} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.2

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.3.1.2.1

Factor $x$ out of $- 16 x$ .

$a = 4 + \frac{x \cdot - 16}{x^{2}} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.2.2

Cancel the common factors.

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Step 4.3.1.2.2.1

Factor $x$ out of $x^{2}$ .

$a = 4 + \frac{x \cdot - 16}{x \cdot x} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.2.2.2

Cancel the common factor.

$a = 4 + \frac{x \cdot - 16}{x \cdot x} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.2.2.3

Rewrite the expression.

$a = 4 + \frac{- 16}{x} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.3

Move the negative in front of the fraction.

$a = 4 - \frac{16}{x} + \frac{9}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.4

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.3.1.4.1

Factor $x$ out of $- b x$ .

$a = 4 - \frac{16}{x} + \frac{9}{x^{2}} + \frac{x (- b)}{x^{2}} + \frac{- c}{x^{2}}$

Step 4.3.1.4.2

Cancel the common factors.

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Step 4.3.1.4.2.1

Factor $x$ out of $x^{2}$ .

$a = 4 - \frac{16}{x} + \frac{9}{x^{2}} + \frac{x (- b)}{x \cdot x} + \frac{- c}{x^{2}}$

Step 4.3.1.4.2.2

Cancel the common factor.

$a = 4 - \frac{16}{x} + \frac{9}{x^{2}} + \frac{x (- b)}{x \cdot x} + \frac{- c}{x^{2}}$

Step 4.3.1.4.2.3

Rewrite the expression.

$a = 4 - \frac{16}{x} + \frac{9}{x^{2}} + \frac{- b}{x} + \frac{- c}{x^{2}}$

Step 4.3.1.5

Move the negative in front of the fraction.

$a = 4 - \frac{16}{x} + \frac{9}{x^{2}} - \frac{b}{x} + \frac{- c}{x^{2}}$

Step 4.3.1.6

Move the negative in front of the fraction.

$a = 4 - \frac{16}{x} + \frac{9}{x^{2}} - \frac{b}{x} - \frac{c}{x^{2}}$