Linear Algebra Examples

$16 x^{2} + 25 y^{2} + 32 x^{2} - 100 y - 284 = 0$

Step 1

Add $16 x^{2}$ and $32 x^{2}$ .

$48 x^{2} + 25 y^{2} - 100 y - 284 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 25$ , $b = - 100$ , and $c = 48 x^{2} - 284$ into the quadratic formula and solve for $y$ .

$\frac{100 \pm \sqrt{{(- 100)}^{2} - 4 \cdot (25 \cdot (48 x^{2} - 284))}}{2 \cdot 25}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 100$ to the power of $2$ .

$y = \frac{100 \pm \sqrt{10000 - 4 \cdot 25 \cdot (48 x^{2} - 284)}}{2 \cdot 25}$

Step 4.1.2

Multiply $- 4$ by $25$ .

$y = \frac{100 \pm \sqrt{10000 - 100 \cdot (48 x^{2} - 284)}}{2 \cdot 25}$

Step 4.1.3

Apply the distributive property.

$y = \frac{100 \pm \sqrt{10000 - 100 (48 x^{2}) - 100 \cdot - 284}}{2 \cdot 25}$

Step 4.1.4

Multiply $48$ by $- 100$ .

$y = \frac{100 \pm \sqrt{10000 - 4800 x^{2} - 100 \cdot - 284}}{2 \cdot 25}$

Step 4.1.5

Multiply $- 100$ by $- 284$ .

$y = \frac{100 \pm \sqrt{10000 - 4800 x^{2} + 28400}}{2 \cdot 25}$

Step 4.1.6

Add $10000$ and $28400$ .

$y = \frac{100 \pm \sqrt{- 4800 x^{2} + 38400}}{2 \cdot 25}$

Step 4.1.7

Factor $4800$ out of $- 4800 x^{2} + 38400$ .

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Step 4.1.7.1

Factor $4800$ out of $- 4800 x^{2}$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2}) + 38400}}{2 \cdot 25}$

Step 4.1.7.2

Factor $4800$ out of $38400$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2}) + 4800 (8)}}{2 \cdot 25}$

Step 4.1.7.3

Factor $4800$ out of $4800 (- x^{2}) + 4800 (8)$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2} + 8)}}{2 \cdot 25}$

Step 4.1.8

Rewrite $4800 (- x^{2} + 8)$ as $40^{2} \cdot (3 (- x^{2} + 8))$ .

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Step 4.1.8.1

Factor $1600$ out of $4800$ .

$y = \frac{100 \pm \sqrt{1600 (3) (- x^{2} + 8)}}{2 \cdot 25}$

Step 4.1.8.2

Rewrite $1600$ as $40^{2}$ .

$y = \frac{100 \pm \sqrt{40^{2} \cdot (3 (- x^{2} + 8))}}{2 \cdot 25}$

Step 4.1.8.3

Add parentheses.

$y = \frac{100 \pm \sqrt{40^{2} \cdot (3 (- x^{2} + 8))}}{2 \cdot 25}$

Step 4.1.9

Pull terms out from under the radical.

$y = \frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{2 \cdot 25}$

Step 4.2

Multiply $2$ by $25$ .

$y = \frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{50}$

Step 4.3

Simplify $\frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{50}$ .

$y = \frac{10 \pm 4 \sqrt{3 (- x^{2} + 8)}}{5}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 100$ to the power of $2$ .

$y = \frac{100 \pm \sqrt{10000 - 4 \cdot 25 \cdot (48 x^{2} - 284)}}{2 \cdot 25}$

Step 5.1.2

Multiply $- 4$ by $25$ .

$y = \frac{100 \pm \sqrt{10000 - 100 \cdot (48 x^{2} - 284)}}{2 \cdot 25}$

Step 5.1.3

Apply the distributive property.

$y = \frac{100 \pm \sqrt{10000 - 100 (48 x^{2}) - 100 \cdot - 284}}{2 \cdot 25}$

Step 5.1.4

Multiply $48$ by $- 100$ .

$y = \frac{100 \pm \sqrt{10000 - 4800 x^{2} - 100 \cdot - 284}}{2 \cdot 25}$

Step 5.1.5

Multiply $- 100$ by $- 284$ .

$y = \frac{100 \pm \sqrt{10000 - 4800 x^{2} + 28400}}{2 \cdot 25}$

Step 5.1.6

Add $10000$ and $28400$ .

$y = \frac{100 \pm \sqrt{- 4800 x^{2} + 38400}}{2 \cdot 25}$

Step 5.1.7

Factor $4800$ out of $- 4800 x^{2} + 38400$ .

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Step 5.1.7.1

Factor $4800$ out of $- 4800 x^{2}$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2}) + 38400}}{2 \cdot 25}$

Step 5.1.7.2

Factor $4800$ out of $38400$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2}) + 4800 (8)}}{2 \cdot 25}$

Step 5.1.7.3

Factor $4800$ out of $4800 (- x^{2}) + 4800 (8)$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2} + 8)}}{2 \cdot 25}$

Step 5.1.8

Rewrite $4800 (- x^{2} + 8)$ as $40^{2} \cdot (3 (- x^{2} + 8))$ .

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Step 5.1.8.1

Factor $1600$ out of $4800$ .

$y = \frac{100 \pm \sqrt{1600 (3) (- x^{2} + 8)}}{2 \cdot 25}$

Step 5.1.8.2

Rewrite $1600$ as $40^{2}$ .

$y = \frac{100 \pm \sqrt{40^{2} \cdot (3 (- x^{2} + 8))}}{2 \cdot 25}$

Step 5.1.8.3

Add parentheses.

$y = \frac{100 \pm \sqrt{40^{2} \cdot (3 (- x^{2} + 8))}}{2 \cdot 25}$

Step 5.1.9

Pull terms out from under the radical.

$y = \frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{2 \cdot 25}$

Step 5.2

Multiply $2$ by $25$ .

$y = \frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{50}$

Step 5.3

Simplify $\frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{50}$ .

$y = \frac{10 \pm 4 \sqrt{3 (- x^{2} + 8)}}{5}$

Step 5.4

Change the $\pm$ to $+$ .

$y = \frac{10 + 4 \sqrt{3 (- x^{2} + 8)}}{5}$

Step 5.5

Factor $2$ out of $10 + 4 \sqrt{3 (- x^{2} + 8)}$ .

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Step 5.5.1

Factor $2$ out of $10$ .

$y = \frac{2 (5) + 4 \sqrt{3 (- x^{2} + 8)}}{5}$

Step 5.5.2

Factor $2$ out of $4 \sqrt{3 (- x^{2} + 8)}$ .

$y = \frac{2 (5) + 2 (2 \sqrt{3 (- x^{2} + 8)})}{5}$

Step 5.5.3

Factor $2$ out of $2 (5) + 2 (2 \sqrt{3 (- x^{2} + 8)})$ .

$y = \frac{2 (5 + 2 \sqrt{3 (- x^{2} + 8)})}{5}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 100$ to the power of $2$ .

$y = \frac{100 \pm \sqrt{10000 - 4 \cdot 25 \cdot (48 x^{2} - 284)}}{2 \cdot 25}$

Step 6.1.2

Multiply $- 4$ by $25$ .

$y = \frac{100 \pm \sqrt{10000 - 100 \cdot (48 x^{2} - 284)}}{2 \cdot 25}$

Step 6.1.3

Apply the distributive property.

$y = \frac{100 \pm \sqrt{10000 - 100 (48 x^{2}) - 100 \cdot - 284}}{2 \cdot 25}$

Step 6.1.4

Multiply $48$ by $- 100$ .

$y = \frac{100 \pm \sqrt{10000 - 4800 x^{2} - 100 \cdot - 284}}{2 \cdot 25}$

Step 6.1.5

Multiply $- 100$ by $- 284$ .

$y = \frac{100 \pm \sqrt{10000 - 4800 x^{2} + 28400}}{2 \cdot 25}$

Step 6.1.6

Add $10000$ and $28400$ .

$y = \frac{100 \pm \sqrt{- 4800 x^{2} + 38400}}{2 \cdot 25}$

Step 6.1.7

Factor $4800$ out of $- 4800 x^{2} + 38400$ .

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Step 6.1.7.1

Factor $4800$ out of $- 4800 x^{2}$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2}) + 38400}}{2 \cdot 25}$

Step 6.1.7.2

Factor $4800$ out of $38400$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2}) + 4800 (8)}}{2 \cdot 25}$

Step 6.1.7.3

Factor $4800$ out of $4800 (- x^{2}) + 4800 (8)$ .

$y = \frac{100 \pm \sqrt{4800 (- x^{2} + 8)}}{2 \cdot 25}$

Step 6.1.8

Rewrite $4800 (- x^{2} + 8)$ as $40^{2} \cdot (3 (- x^{2} + 8))$ .

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Step 6.1.8.1

Factor $1600$ out of $4800$ .

$y = \frac{100 \pm \sqrt{1600 (3) (- x^{2} + 8)}}{2 \cdot 25}$

Step 6.1.8.2

Rewrite $1600$ as $40^{2}$ .

$y = \frac{100 \pm \sqrt{40^{2} \cdot (3 (- x^{2} + 8))}}{2 \cdot 25}$

Step 6.1.8.3

Add parentheses.

$y = \frac{100 \pm \sqrt{40^{2} \cdot (3 (- x^{2} + 8))}}{2 \cdot 25}$

Step 6.1.9

Pull terms out from under the radical.

$y = \frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{2 \cdot 25}$

Step 6.2

Multiply $2$ by $25$ .

$y = \frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{50}$

Step 6.3

Simplify $\frac{100 \pm 40 \sqrt{3 (- x^{2} + 8)}}{50}$ .

$y = \frac{10 \pm 4 \sqrt{3 (- x^{2} + 8)}}{5}$

Step 6.4

Change the $\pm$ to $-$ .

$y = \frac{10 - 4 \sqrt{3 (- x^{2} + 8)}}{5}$

Step 6.5

Factor $2$ out of $10 - 4 \sqrt{3 (- x^{2} + 8)}$ .

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Step 6.5.1

Factor $2$ out of $10$ .

$y = \frac{2 (5) - 4 \sqrt{3 (- x^{2} + 8)}}{5}$

Step 6.5.2

Factor $2$ out of $- 4 \sqrt{3 (- x^{2} + 8)}$ .

$y = \frac{2 (5) + 2 (- 2 \sqrt{3 (- x^{2} + 8)})}{5}$

Step 6.5.3

Factor $2$ out of $2 (5) + 2 (- 2 \sqrt{3 (- x^{2} + 8)})$ .

$y = \frac{2 (5 - 2 \sqrt{3 (- x^{2} + 8)})}{5}$

Step 7

The final answer is the combination of both solutions.

$y = \frac{2 (5 + 2 \sqrt{3 (- x^{2} + 8)})}{5}$

$y = \frac{2 (5 - 2 \sqrt{3 (- x^{2} + 8)})}{5}$

Step 8

Set the radicand in $\sqrt{3 (- x^{2} + 8)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (- x^{2} + 8) \geq 0$

Step 9

Solve for $x$ .

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Step 9.1

Divide each term in $3 (- x^{2} + 8) \geq 0$ by $3$ and simplify.

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Step 9.1.1

Divide each term in $3 (- x^{2} + 8) \geq 0$ by $3$ .

$\frac{3 (- x^{2} + 8)}{3} \geq \frac{0}{3}$

Step 9.1.2

Simplify the left side.

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Step 9.1.2.1

Cancel the common factor of $3$ .

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Step 9.1.2.1.1

Cancel the common factor.

$\frac{3 (- x^{2} + 8)}{3} \geq \frac{0}{3}$

Step 9.1.2.1.2

Divide $- x^{2} + 8$ by $1$ .

$- x^{2} + 8 \geq \frac{0}{3}$

Step 9.1.3

Simplify the right side.

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Step 9.1.3.1

Divide $0$ by $3$ .

$- x^{2} + 8 \geq 0$

Step 9.2

Subtract $8$ from both sides of the inequality.

$- x^{2} \geq - 8$

Step 9.3

Divide each term in $- x^{2} \geq - 8$ by $- 1$ and simplify.

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Step 9.3.1

Divide each term in $- x^{2} \geq - 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 8}{- 1}$

Step 9.3.2

Simplify the left side.

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Step 9.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 8}{- 1}$

Step 9.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 8}{- 1}$

Step 9.3.3

Simplify the right side.

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Step 9.3.3.1

Divide $- 8$ by $- 1$ .

$x^{2} \leq 8$

Step 9.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{8}$

Step 9.5

Simplify the equation.

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Step 9.5.1

Simplify the left side.

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Step 9.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{8}$

Step 9.5.2

Simplify the right side.

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Step 9.5.2.1

Simplify $\sqrt{8}$ .

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Step 9.5.2.1.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 9.5.2.1.1.1

Factor $4$ out of $8$ .

$| x | \leq \sqrt{4 (2)}$

Step 9.5.2.1.1.2

Rewrite $4$ as $2^{2}$ .

$| x | \leq \sqrt{2^{2} \cdot 2}$

Step 9.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 2 | \sqrt{2}$

Step 9.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x | \leq 2 \sqrt{2}$

Step 9.6

Write $| x | \leq 2 \sqrt{2}$ as a piecewise.

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Step 9.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 9.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 2 \sqrt{2}$

Step 9.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 9.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 2 \sqrt{2}$

Step 9.6.5

Write as a piecewise.

${\begin{cases} x \leq 2 \sqrt{2} & x \geq 0 \\ - x \leq 2 \sqrt{2} & x < 0 \end{cases}$

Step 9.7

Find the intersection of $x \leq 2 \sqrt{2}$ and $x \geq 0$ .

$0 \leq x \leq 2 \sqrt{2}$

Step 9.8

Solve $- x \leq 2 \sqrt{2}$ when $x < 0$ .

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Step 9.8.1

Divide each term in $- x \leq 2 \sqrt{2}$ by $- 1$ and simplify.

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Step 9.8.1.1

Divide each term in $- x \leq 2 \sqrt{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{2 \sqrt{2}}{- 1}$

Step 9.8.1.2

Simplify the left side.

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Step 9.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{2 \sqrt{2}}{- 1}$

Step 9.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{2 \sqrt{2}}{- 1}$

Step 9.8.1.3

Simplify the right side.

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Step 9.8.1.3.1

Move the negative one from the denominator of $\frac{2 \sqrt{2}}{- 1}$ .

$x \geq - 1 \cdot (2 \sqrt{2})$

Step 9.8.1.3.2

Rewrite $- 1 \cdot (2 \sqrt{2})$ as $- (2 \sqrt{2})$ .

$x \geq - (2 \sqrt{2})$

Step 9.8.1.3.3

Multiply $2$ by $- 1$ .

$x \geq - 2 \sqrt{2}$

Step 9.8.2

Find the intersection of $x \geq - 2 \sqrt{2}$ and $x < 0$ .

$- 2 \sqrt{2} \leq x < 0$

Step 9.9

Find the union of the solutions.

$- 2 \sqrt{2} \leq x \leq 2 \sqrt{2}$

Step 10

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 2 \sqrt{2}, 2 \sqrt{2}]$

Set-Builder Notation:

${x | - 2 \sqrt{2} \leq x \leq 2 \sqrt{2}}$

Step 11