Algebra Examples

$a c + 2 b c - 6 a b - 3 a^{2} = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = - 3$ , $b = c - 6 b$ , and $c = 2 b c$ into the quadratic formula and solve for $a$ .

$\frac{- (c - 6 b) \pm \sqrt{{(c - 6 b)}^{2} - 4 \cdot (- 3 \cdot (2 b c))}}{2 \cdot - 3}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Apply the distributive property.

$a = \frac{- c - (- 6 b) \pm \sqrt{{(c - 6 b)}^{2} - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.2

Multiply $- 6$ by $- 1$ .

$a = \frac{- c + 6 b \pm \sqrt{{(c - 6 b)}^{2} - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.3

Rewrite ${(c - 6 b)}^{2}$ as $(c - 6 b) (c - 6 b)$ .

$a = \frac{- c + 6 b \pm \sqrt{(c - 6 b) (c - 6 b) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.4

Expand $(c - 6 b) (c - 6 b)$ using the FOIL Method.

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Step 3.1.4.1

Apply the distributive property.

$a = \frac{- c + 6 b \pm \sqrt{c (c - 6 b) - 6 b (c - 6 b) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.4.2

Apply the distributive property.

$a = \frac{- c + 6 b \pm \sqrt{c \cdot c + c (- 6 b) - 6 b (c - 6 b) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.4.3

Apply the distributive property.

$a = \frac{- c + 6 b \pm \sqrt{c \cdot c + c (- 6 b) - 6 b c - 6 b (- 6 b) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5

Simplify and combine like terms.

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Step 3.1.5.1

Simplify each term.

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Step 3.1.5.1.1

Multiply $c$ by $c$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} + c (- 6 b) - 6 b c - 6 b (- 6 b) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5.1.2

Rewrite using the commutative property of multiplication.

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 6 c b - 6 b c - 6 b (- 6 b) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5.1.3

Rewrite using the commutative property of multiplication.

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 6 c b - 6 b c - 6 \cdot (- 6 b \cdot b) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5.1.4

Multiply $b$ by $b$ by adding the exponents.

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Step 3.1.5.1.4.1

Move $b$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 6 c b - 6 b c - 6 \cdot (- 6 (b \cdot b)) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5.1.4.2

Multiply $b$ by $b$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 6 c b - 6 b c - 6 \cdot (- 6 b^{2}) - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5.1.5

Multiply $- 6$ by $- 6$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 6 c b - 6 b c + 36 b^{2} - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5.2

Subtract $6 b c$ from $- 6 c b$ .

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Step 3.1.5.2.1

Move $c$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 6 b c - 6 b c + 36 b^{2} - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.5.2.2

Subtract $6 b c$ from $- 6 b c$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 12 b c + 36 b^{2} - 4 \cdot - 3 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.6

Multiply $- 4 \cdot - 3 \cdot 2$ .

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Step 3.1.6.1

Multiply $- 4$ by $- 3$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 12 b c + 36 b^{2} + 12 \cdot (2 b c)}}{2 \cdot - 3}$

Step 3.1.6.2

Multiply $12$ by $2$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} - 12 b c + 36 b^{2} + 24 b c}}{2 \cdot - 3}$

Step 3.1.7

Add $- 12 b c$ and $24 b c$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} + 36 b^{2} + 12 b c}}{2 \cdot - 3}$

Step 3.1.8

Factor using the perfect square rule.

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Step 3.1.8.1

Rearrange terms.

$a = \frac{- c + 6 b \pm \sqrt{c^{2} + 12 b c + 36 b^{2}}}{2 \cdot - 3}$

Step 3.1.8.2

Rewrite $36 b^{2}$ as ${(6 b)}^{2}$ .

$a = \frac{- c + 6 b \pm \sqrt{c^{2} + 12 b c + {(6 b)}^{2}}}{2 \cdot - 3}$

Step 3.1.8.3

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

$12 b c = 2 \cdot c \cdot (6 b)$

Step 3.1.8.4

Rewrite the polynomial.

$a = \frac{- c + 6 b \pm \sqrt{c^{2} + 2 \cdot c \cdot (6 b) + {(6 b)}^{2}}}{2 \cdot - 3}$

Step 3.1.8.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = c$ and $b = 6 b$ .

$a = \frac{- c + 6 b \pm \sqrt{{(c + 6 b)}^{2}}}{2 \cdot - 3}$

Step 3.1.9

Pull terms out from under the radical, assuming positive real numbers.

$a = \frac{- c + 6 b \pm (c + 6 b)}{2 \cdot - 3}$

Step 3.2

Multiply $2$ by $- 3$ .

$a = \frac{- c + 6 b \pm (c + 6 b)}{- 6}$

Step 3.3

Simplify $\frac{- c + 6 b \pm (c + 6 b)}{- 6}$ .

$a = \frac{c - 6 b \pm (- (- c - 6 b))}{6}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Apply the distributive property.

$a = \frac{c - 6 b \pm (c - (- 6 b))}{6}$

Step 3.4.2

Multiply $- - c$ .

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Step 3.4.2.1

Multiply $- 1$ by $- 1$ .

$a = \frac{c - 6 b \pm (1 c - (- 6 b))}{6}$

Step 3.4.2.2

Multiply $c$ by $1$ .

$a = \frac{c - 6 b \pm (c - (- 6 b))}{6}$

Step 3.4.3

Multiply $- 6$ by $- 1$ .

$a = \frac{c - 6 b \pm (c + 6 b)}{6}$

Step 4

The final answer is the combination of both solutions.

$a = \frac{c}{3}$

$a = - 2 b$