Calculus Examples

${(r - 3)}^{2} = (r^{2} - 5 r) + 3$

Step 1

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

$r^{2} - 5 r + 3 = {(r - 3)}^{2}$

Step 2

Simplify ${(r - 3)}^{2}$ .

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Step 2.1

Rewrite.

$r^{2} - 5 r + 3 = 0 + 0 + {(r - 3)}^{2}$

Step 2.2

Rewrite ${(r - 3)}^{2}$ as $(r - 3) (r - 3)$ .

$r^{2} - 5 r + 3 = (r - 3) (r - 3)$

Step 2.3

Expand $(r - 3) (r - 3)$ using the FOIL Method.

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Step 2.3.1

Apply the distributive property.

$r^{2} - 5 r + 3 = r (r - 3) - 3 (r - 3)$

Step 2.3.2

Apply the distributive property.

$r^{2} - 5 r + 3 = r \cdot r + r \cdot - 3 - 3 (r - 3)$

Step 2.3.3

Apply the distributive property.

$r^{2} - 5 r + 3 = r \cdot r + r \cdot - 3 - 3 r - 3 \cdot - 3$

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 $r$ by $r$ .

$r^{2} - 5 r + 3 = r^{2} + r \cdot - 3 - 3 r - 3 \cdot - 3$

Step 2.4.1.2

Move $- 3$ to the left of $r$ .

$r^{2} - 5 r + 3 = r^{2} - 3 \cdot r - 3 r - 3 \cdot - 3$

Step 2.4.1.3

Multiply $- 3$ by $- 3$ .

$r^{2} - 5 r + 3 = r^{2} - 3 r - 3 r + 9$

Step 2.4.2

Subtract $3 r$ from $- 3 r$ .

$r^{2} - 5 r + 3 = r^{2} - 6 r + 9$

Step 3

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

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Step 3.1

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

$r^{2} - 5 r + 3 - r^{2} = - 6 r + 9$

Step 3.2

Add $6 r$ to both sides of the equation.

$r^{2} - 5 r + 3 - r^{2} + 6 r = 9$

Step 3.3

Combine the opposite terms in $r^{2} - 5 r + 3 - r^{2} + 6 r$ .

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Step 3.3.1

Subtract $r^{2}$ from $r^{2}$ .

$- 5 r + 3 + 0 + 6 r = 9$

Step 3.3.2

Add $- 5 r + 3$ and $0$ .

$- 5 r + 3 + 6 r = 9$

Step 3.4

Add $- 5 r$ and $6 r$ .

$r + 3 = 9$

Step 4

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

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Step 4.1

Subtract $3$ from both sides of the equation.

$r = 9 - 3$

Step 4.2

Subtract $3$ from $9$ .

$r = 6$