Algebra Examples

$12 x^{2} + 12 y^{2} - 84 y + 147 = 0$

Step 1

Solve for $y$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 12$ , $b = - 84$ , and $c = 12 x^{2} + 147$ into the quadratic formula and solve for $y$ .

$\frac{84 \pm \sqrt{{(- 84)}^{2} - 4 \cdot (12 \cdot (12 x^{2} + 147))}}{2 \cdot 12}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Raise $- 84$ to the power of $2$ .

$y = \frac{84 \pm \sqrt{7056 - 4 \cdot 12 \cdot (12 x^{2} + 147)}}{2 \cdot 12}$

Step 1.3.1.2

Multiply $- 4$ by $12$ .

$y = \frac{84 \pm \sqrt{7056 - 48 \cdot (12 x^{2} + 147)}}{2 \cdot 12}$

Step 1.3.1.3

Apply the distributive property.

$y = \frac{84 \pm \sqrt{7056 - 48 (12 x^{2}) - 48 \cdot 147}}{2 \cdot 12}$

Step 1.3.1.4

Multiply $12$ by $- 48$ .

$y = \frac{84 \pm \sqrt{7056 - 576 x^{2} - 48 \cdot 147}}{2 \cdot 12}$

Step 1.3.1.5

Multiply $- 48$ by $147$ .

$y = \frac{84 \pm \sqrt{7056 - 576 x^{2} - 7056}}{2 \cdot 12}$

Step 1.3.1.6

Subtract $7056$ from $7056$ .

$y = \frac{84 \pm \sqrt{- 576 x^{2} + 0}}{2 \cdot 12}$

Step 1.3.1.7

Add $- 576 x^{2}$ and $0$ .

$y = \frac{84 \pm \sqrt{- 576 x^{2}}}{2 \cdot 12}$

Step 1.3.1.8

Rewrite $- 576 x^{2}$ as ${(24 x)}^{2} \cdot - 1$ .

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Step 1.3.1.8.1

Factor $576$ out of $- 576$ .

$y = \frac{84 \pm \sqrt{576 (- 1) x^{2}}}{2 \cdot 12}$

Step 1.3.1.8.2

Rewrite $576$ as $24^{2}$ .

$y = \frac{84 \pm \sqrt{24^{2} \cdot (- 1 x^{2})}}{2 \cdot 12}$

Step 1.3.1.8.3

Move $- 1$ .

$y = \frac{84 \pm \sqrt{24^{2} x^{2} \cdot - 1}}{2 \cdot 12}$

Step 1.3.1.8.4

Rewrite $24^{2} x^{2}$ as ${(24 x)}^{2}$ .

$y = \frac{84 \pm \sqrt{{(24 x)}^{2} \cdot - 1}}{2 \cdot 12}$

Step 1.3.1.9

Pull terms out from under the radical.

$y = \frac{84 \pm 24 x \sqrt{- 1}}{2 \cdot 12}$

Step 1.3.1.10

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{84 \pm 24 x i}{2 \cdot 12}$

Step 1.3.2

Multiply $2$ by $12$ .

$y = \frac{84 \pm 24 x i}{24}$

Step 1.3.3

Simplify $\frac{84 \pm 24 x i}{24}$ .

$y = \frac{7 \pm 2 x i}{2}$

Step 1.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $- 84$ to the power of $2$ .

$y = \frac{84 \pm \sqrt{7056 - 4 \cdot 12 \cdot (12 x^{2} + 147)}}{2 \cdot 12}$

Step 1.4.1.2

Multiply $- 4$ by $12$ .

$y = \frac{84 \pm \sqrt{7056 - 48 \cdot (12 x^{2} + 147)}}{2 \cdot 12}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{84 \pm \sqrt{7056 - 48 (12 x^{2}) - 48 \cdot 147}}{2 \cdot 12}$

Step 1.4.1.4

Multiply $12$ by $- 48$ .

$y = \frac{84 \pm \sqrt{7056 - 576 x^{2} - 48 \cdot 147}}{2 \cdot 12}$

Step 1.4.1.5

Multiply $- 48$ by $147$ .

$y = \frac{84 \pm \sqrt{7056 - 576 x^{2} - 7056}}{2 \cdot 12}$

Step 1.4.1.6

Subtract $7056$ from $7056$ .

$y = \frac{84 \pm \sqrt{- 576 x^{2} + 0}}{2 \cdot 12}$

Step 1.4.1.7

Add $- 576 x^{2}$ and $0$ .

$y = \frac{84 \pm \sqrt{- 576 x^{2}}}{2 \cdot 12}$

Step 1.4.1.8

Rewrite $- 576 x^{2}$ as ${(24 x)}^{2} \cdot - 1$ .

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Step 1.4.1.8.1

Factor $576$ out of $- 576$ .

$y = \frac{84 \pm \sqrt{576 (- 1) x^{2}}}{2 \cdot 12}$

Step 1.4.1.8.2

Rewrite $576$ as $24^{2}$ .

$y = \frac{84 \pm \sqrt{24^{2} \cdot (- 1 x^{2})}}{2 \cdot 12}$

Step 1.4.1.8.3

Move $- 1$ .

$y = \frac{84 \pm \sqrt{24^{2} x^{2} \cdot - 1}}{2 \cdot 12}$

Step 1.4.1.8.4

Rewrite $24^{2} x^{2}$ as ${(24 x)}^{2}$ .

$y = \frac{84 \pm \sqrt{{(24 x)}^{2} \cdot - 1}}{2 \cdot 12}$

Step 1.4.1.9

Pull terms out from under the radical.

$y = \frac{84 \pm 24 x \sqrt{- 1}}{2 \cdot 12}$

Step 1.4.1.10

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{84 \pm 24 x i}{2 \cdot 12}$

Step 1.4.2

Multiply $2$ by $12$ .

$y = \frac{84 \pm 24 x i}{24}$

Step 1.4.3

Simplify $\frac{84 \pm 24 x i}{24}$ .

$y = \frac{7 \pm 2 x i}{2}$

Step 1.4.4

Change the $\pm$ to $+$ .

$y = \frac{7 + 2 x i}{2}$

Step 1.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $- 84$ to the power of $2$ .

$y = \frac{84 \pm \sqrt{7056 - 4 \cdot 12 \cdot (12 x^{2} + 147)}}{2 \cdot 12}$

Step 1.5.1.2

Multiply $- 4$ by $12$ .

$y = \frac{84 \pm \sqrt{7056 - 48 \cdot (12 x^{2} + 147)}}{2 \cdot 12}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{84 \pm \sqrt{7056 - 48 (12 x^{2}) - 48 \cdot 147}}{2 \cdot 12}$

Step 1.5.1.4

Multiply $12$ by $- 48$ .

$y = \frac{84 \pm \sqrt{7056 - 576 x^{2} - 48 \cdot 147}}{2 \cdot 12}$

Step 1.5.1.5

Multiply $- 48$ by $147$ .

$y = \frac{84 \pm \sqrt{7056 - 576 x^{2} - 7056}}{2 \cdot 12}$

Step 1.5.1.6

Subtract $7056$ from $7056$ .

$y = \frac{84 \pm \sqrt{- 576 x^{2} + 0}}{2 \cdot 12}$

Step 1.5.1.7

Add $- 576 x^{2}$ and $0$ .

$y = \frac{84 \pm \sqrt{- 576 x^{2}}}{2 \cdot 12}$

Step 1.5.1.8

Rewrite $- 576 x^{2}$ as ${(24 x)}^{2} \cdot - 1$ .

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Step 1.5.1.8.1

Factor $576$ out of $- 576$ .

$y = \frac{84 \pm \sqrt{576 (- 1) x^{2}}}{2 \cdot 12}$

Step 1.5.1.8.2

Rewrite $576$ as $24^{2}$ .

$y = \frac{84 \pm \sqrt{24^{2} \cdot (- 1 x^{2})}}{2 \cdot 12}$

Step 1.5.1.8.3

Move $- 1$ .

$y = \frac{84 \pm \sqrt{24^{2} x^{2} \cdot - 1}}{2 \cdot 12}$

Step 1.5.1.8.4

Rewrite $24^{2} x^{2}$ as ${(24 x)}^{2}$ .

$y = \frac{84 \pm \sqrt{{(24 x)}^{2} \cdot - 1}}{2 \cdot 12}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{84 \pm 24 x \sqrt{- 1}}{2 \cdot 12}$

Step 1.5.1.10

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{84 \pm 24 x i}{2 \cdot 12}$

Step 1.5.2

Multiply $2$ by $12$ .

$y = \frac{84 \pm 24 x i}{24}$

Step 1.5.3

Simplify $\frac{84 \pm 24 x i}{24}$ .

$y = \frac{7 \pm 2 x i}{2}$

Step 1.5.4

Change the $\pm$ to $-$ .

$y = \frac{7 - 2 x i}{2}$

Step 1.6

The final answer is the combination of both solutions.

$y = \frac{7 + 2 x i}{2}$

$y = \frac{7 - 2 x i}{2}$

$y = \frac{7 + 2 x i}{2}$

$y = \frac{7 - 2 x i}{2}$

Step 2

Divide the first expression by the second expression.

$y = \frac{\frac{7 + 2 x i}{2}}{\frac{7 - 2 x i}{2}}$

Step 3

Expand $\frac{7 + 2 x i}{2}$ .

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Step 3.1

Split the fraction $\frac{7 + 2 x i}{2}$ into two fractions.

$\frac{\frac{7}{2} + \frac{2 x i}{2}}{\frac{7 - 2 x i}{2}}$

Step 3.2

Reorder $\frac{7}{2}$ and $\frac{2 x i}{2}$ .

$\frac{\frac{2 x i}{2} + \frac{7}{2}}{\frac{7 - 2 x i}{2}}$

Step 4

Expand $\frac{7 - 2 x i}{2}$ .

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Step 4.1

Split the fraction $\frac{7 - 2 x i}{2}$ into two fractions.

$\frac{\frac{2 x i}{2} + \frac{7}{2}}{\frac{7}{2} + \frac{- 2 x i}{2}}$

Step 4.2

Reorder $\frac{7}{2}$ and $\frac{- 2 x i}{2}$ .

$\frac{\frac{2 x i}{2} + \frac{7}{2}}{\frac{- 2 x i}{2} + \frac{7}{2}}$

Step 5

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$\frac{- 2 x i}{2}$

$\frac{7}{2}$

$\frac{2 x i}{2}$

$\frac{7}{2}$

Step 6

Divide the highest order term in the dividend $\frac{2 x i}{2}$ by the highest order term in divisor $\frac{- 2 x i}{2}$ .

				-	$1$
	$\frac{- 2 x i}{2}$	+	$\frac{7}{2}$		$\frac{2 x i}{2}$	+	$\frac{7}{2}$

Step 7

Multiply the new quotient term by the divisor.

			-	$1$
$\frac{- 2 x i}{2}$	+	$\frac{7}{2}$		$\frac{2 x i}{2}$	+	$\frac{7}{2}$
			+	$x i$	-	$\frac{7}{2}$

Step 8

The expression needs to be subtracted from the dividend, so change all the signs in $x i - \frac{7}{2}$

			-	$1$
$\frac{- 2 x i}{2}$	+	$\frac{7}{2}$		$\frac{2 x i}{2}$	+	$\frac{7}{2}$
			-	$x i$	+	$\frac{7}{2}$

Step 9

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

			-	$1$
$\frac{- 2 x i}{2}$	+	$\frac{7}{2}$		$\frac{2 x i}{2}$	+	$\frac{7}{2}$
			-	$x i$	+	$\frac{7}{2}$
					+	$7$

Step 10

The final answer is the quotient plus the remainder over the divisor.

$y = - 1 + \frac{7}{\frac{- 2 x i}{2} + \frac{7}{2}}$

Step 11