Algebra Examples

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

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

Step 1

Use the quadratic formula to find the solutions.

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

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

Step 2

Substitute the values

a = - 3

$a = - 3$ ,

b = c - 6 b

$b = c - 6 b$ , and

c = 2 b c

$c = 2 b c$ into the quadratic formula and solve for

a

$a$ .

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

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

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}

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

Step 3.1.2

Multiply

- 6

$- 6$ by

- 1

$- 1$ .

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

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

Step 3.1.3

Rewrite

{(c - 6 b)}^{2}

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

(c - 6 b) (c - 6 b)

$(c - 6 b) (c - 6 b)$ .

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

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

Step 3.1.4

Expand

(c - 6 b) (c - 6 b)

$(c - 6 b) (c - 6 b)$ 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}

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

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}

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

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}

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

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}

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

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

$c$ by

c

$c$ .

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}

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

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}

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

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}

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

Step 3.1.5.1.4

Multiply

b

$b$ by

b

$b$ by adding the exponents.

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Step 3.1.5.1.4.1

Move

b

$b$ .

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}

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

Step 3.1.5.1.4.2

Multiply

b

$b$ by

b

$b$ .

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}

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

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.

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

$a = - 2 b$