Pre-Algebra Examples

$- 7 r^{2} + 140 r - 700 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $- 7$ out of $- 7 r^{2} + 140 r - 700$ .

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Step 1.1.1

Factor $- 7$ out of $- 7 r^{2}$ .

$- 7 r^{2} + 140 r - 700 = 0$

Step 1.1.2

Factor $- 7$ out of $140 r$ .

$- 7 r^{2} - 7 (- 20 r) - 700 = 0$

Step 1.1.3

Factor $- 7$ out of $- 700$ .

$- 7 r^{2} - 7 (- 20 r) - 7 \cdot 100 = 0$

Step 1.1.4

Factor $- 7$ out of $- 7 (r^{2}) - 7 (- 20 r)$ .

$- 7 (r^{2} - 20 r) - 7 \cdot 100 = 0$

Step 1.1.5

Factor $- 7$ out of $- 7 (r^{2} - 20 r) - 7 (100)$ .

$- 7 (r^{2} - 20 r + 100) = 0$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $100$ as $10^{2}$ .

$- 7 (r^{2} - 20 r + 10^{2}) = 0$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$20 r = 2 \cdot r \cdot 10$

Step 1.2.3

Rewrite the polynomial.

$- 7 (r^{2} - 2 \cdot r \cdot 10 + 10^{2}) = 0$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = r$ and $b = 10$ .

$- 7 {(r - 10)}^{2} = 0$

Step 2

Divide each term in $- 7 {(r - 10)}^{2} = 0$ by $- 7$ and simplify.

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Step 2.1

Divide each term in $- 7 {(r - 10)}^{2} = 0$ by $- 7$ .

$\frac{- 7 {(r - 10)}^{2}}{- 7} = \frac{0}{- 7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 7$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 7 {(r - 10)}^{2}}{- 7} = \frac{0}{- 7}$

Step 2.2.1.2

Divide ${(r - 10)}^{2}$ by $1$ .

${(r - 10)}^{2} = \frac{0}{- 7}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 7$ .

${(r - 10)}^{2} = 0$

Step 3

Set the $r - 10$ equal to $0$ .

$r - 10 = 0$

Step 4

Add $10$ to both sides of the equation.

$r = 10$