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 $c$ to the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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