Linear Algebra Examples

$- 16 x^{2} + y^{2} - 32 x - 14 y + 17 = 0$

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 $a = 1$ , $b = - 14$ , and $c = - 16 x^{2} - 32 x + 17$ into the quadratic formula and solve for $y$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot (- 16 x^{2} - 32 x + 17))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 14$ to the power of $2$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot (- 16 x^{2} - 32 x + 17)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot (- 16 x^{2} - 32 x + 17)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$y = \frac{14 \pm \sqrt{196 - 4 (- 16 x^{2}) - 4 (- 32 x) - 4 \cdot 17}}{2 \cdot 1}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $- 16$ by $- 4$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} - 4 (- 32 x) - 4 \cdot 17}}{2 \cdot 1}$

Step 3.1.4.2

Multiply $- 32$ by $- 4$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} + 128 x - 4 \cdot 17}}{2 \cdot 1}$

Step 3.1.4.3

Multiply $- 4$ by $17$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} + 128 x - 68}}{2 \cdot 1}$

Step 3.1.5

Subtract $68$ from $196$ .

$y = \frac{14 \pm \sqrt{64 x^{2} + 128 x + 128}}{2 \cdot 1}$

Step 3.1.6

Factor $64$ out of $64 x^{2} + 128 x + 128$ .

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Step 3.1.6.1

Factor $64$ out of $64 x^{2}$ .

$y = \frac{14 \pm \sqrt{64 (x^{2}) + 128 x + 128}}{2 \cdot 1}$

Step 3.1.6.2

Factor $64$ out of $128 x$ .

$y = \frac{14 \pm \sqrt{64 (x^{2}) + 64 (2 x) + 128}}{2 \cdot 1}$

Step 3.1.6.3

Factor $64$ out of $128$ .

$y = \frac{14 \pm \sqrt{64 x^{2} + 64 (2 x) + 64 \cdot 2}}{2 \cdot 1}$

Step 3.1.6.4

Factor $64$ out of $64 x^{2} + 64 (2 x)$ .

$y = \frac{14 \pm \sqrt{64 (x^{2} + 2 x) + 64 \cdot 2}}{2 \cdot 1}$

Step 3.1.6.5

Factor $64$ out of $64 (x^{2} + 2 x) + 64 \cdot 2$ .

$y = \frac{14 \pm \sqrt{64 (x^{2} + 2 x + 2)}}{2 \cdot 1}$

Step 3.1.7

Rewrite $64$ as $8^{2}$ .

$y = \frac{14 \pm \sqrt{8^{2} (x^{2} + 2 x + 2)}}{2 \cdot 1}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$y = \frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2}$

Step 3.3

Simplify $\frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2}$ .

$y = 7 \pm 4 \sqrt{x^{2} + 2 x + 2}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 14$ to the power of $2$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot (- 16 x^{2} - 32 x + 17)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot (- 16 x^{2} - 32 x + 17)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$y = \frac{14 \pm \sqrt{196 - 4 (- 16 x^{2}) - 4 (- 32 x) - 4 \cdot 17}}{2 \cdot 1}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $- 16$ by $- 4$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} - 4 (- 32 x) - 4 \cdot 17}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $- 32$ by $- 4$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} + 128 x - 4 \cdot 17}}{2 \cdot 1}$

Step 4.1.4.3

Multiply $- 4$ by $17$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} + 128 x - 68}}{2 \cdot 1}$

Step 4.1.5

Subtract $68$ from $196$ .

$y = \frac{14 \pm \sqrt{64 x^{2} + 128 x + 128}}{2 \cdot 1}$

Step 4.1.6

Factor $64$ out of $64 x^{2} + 128 x + 128$ .

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Step 4.1.6.1

Factor $64$ out of $64 x^{2}$ .

$y = \frac{14 \pm \sqrt{64 (x^{2}) + 128 x + 128}}{2 \cdot 1}$

Step 4.1.6.2

Factor $64$ out of $128 x$ .

$y = \frac{14 \pm \sqrt{64 (x^{2}) + 64 (2 x) + 128}}{2 \cdot 1}$

Step 4.1.6.3

Factor $64$ out of $128$ .

$y = \frac{14 \pm \sqrt{64 x^{2} + 64 (2 x) + 64 \cdot 2}}{2 \cdot 1}$

Step 4.1.6.4

Factor $64$ out of $64 x^{2} + 64 (2 x)$ .

$y = \frac{14 \pm \sqrt{64 (x^{2} + 2 x) + 64 \cdot 2}}{2 \cdot 1}$

Step 4.1.6.5

Factor $64$ out of $64 (x^{2} + 2 x) + 64 \cdot 2$ .

$y = \frac{14 \pm \sqrt{64 (x^{2} + 2 x + 2)}}{2 \cdot 1}$

Step 4.1.7

Rewrite $64$ as $8^{2}$ .

$y = \frac{14 \pm \sqrt{8^{2} (x^{2} + 2 x + 2)}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2}$

Step 4.3

Simplify $\frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2}$ .

$y = 7 \pm 4 \sqrt{x^{2} + 2 x + 2}$

Step 4.4

Change the $\pm$ to $+$ .

$y = 7 + 4 \sqrt{x^{2} + 2 x + 2}$

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 $- 14$ to the power of $2$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot (- 16 x^{2} - 32 x + 17)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot (- 16 x^{2} - 32 x + 17)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$y = \frac{14 \pm \sqrt{196 - 4 (- 16 x^{2}) - 4 (- 32 x) - 4 \cdot 17}}{2 \cdot 1}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $- 16$ by $- 4$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} - 4 (- 32 x) - 4 \cdot 17}}{2 \cdot 1}$

Step 5.1.4.2

Multiply $- 32$ by $- 4$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} + 128 x - 4 \cdot 17}}{2 \cdot 1}$

Step 5.1.4.3

Multiply $- 4$ by $17$ .

$y = \frac{14 \pm \sqrt{196 + 64 x^{2} + 128 x - 68}}{2 \cdot 1}$

Step 5.1.5

Subtract $68$ from $196$ .

$y = \frac{14 \pm \sqrt{64 x^{2} + 128 x + 128}}{2 \cdot 1}$

Step 5.1.6

Factor $64$ out of $64 x^{2} + 128 x + 128$ .

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Step 5.1.6.1

Factor $64$ out of $64 x^{2}$ .

$y = \frac{14 \pm \sqrt{64 (x^{2}) + 128 x + 128}}{2 \cdot 1}$

Step 5.1.6.2

Factor $64$ out of $128 x$ .

$y = \frac{14 \pm \sqrt{64 (x^{2}) + 64 (2 x) + 128}}{2 \cdot 1}$

Step 5.1.6.3

Factor $64$ out of $128$ .

$y = \frac{14 \pm \sqrt{64 x^{2} + 64 (2 x) + 64 \cdot 2}}{2 \cdot 1}$

Step 5.1.6.4

Factor $64$ out of $64 x^{2} + 64 (2 x)$ .

$y = \frac{14 \pm \sqrt{64 (x^{2} + 2 x) + 64 \cdot 2}}{2 \cdot 1}$

Step 5.1.6.5

Factor $64$ out of $64 (x^{2} + 2 x) + 64 \cdot 2$ .

$y = \frac{14 \pm \sqrt{64 (x^{2} + 2 x + 2)}}{2 \cdot 1}$

Step 5.1.7

Rewrite $64$ as $8^{2}$ .

$y = \frac{14 \pm \sqrt{8^{2} (x^{2} + 2 x + 2)}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2}$

Step 5.3

Simplify $\frac{14 \pm 8 \sqrt{x^{2} + 2 x + 2}}{2}$ .

$y = 7 \pm 4 \sqrt{x^{2} + 2 x + 2}$

Step 5.4

Change the $\pm$ to $-$ .

$y = 7 - 4 \sqrt{x^{2} + 2 x + 2}$

Step 6

The final answer is the combination of both solutions.

$y = 7 + 4 \sqrt{x^{2} + 2 x + 2}$

$y = 7 - 4 \sqrt{x^{2} + 2 x + 2}$

Step 7

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

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

Step 8

Solve for $x$ .

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Step 8.1

Convert the inequality to an equation.

$x^{2} + 2 x + 2 = 0$

Step 8.2

Use the quadratic formula to find the solutions.

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

Step 8.3

Substitute the values $a = 1$ , $b = 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 8.4

Simplify.

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Step 8.4.1

Simplify the numerator.

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Step 8.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 8.4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 8.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 8.4.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 8.4.1.3

Subtract $8$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 8.4.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 8.4.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 8.4.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 8.4.1.7

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 8.4.1.8

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

$x = \frac{- 2 \pm i \cdot 2}{2 \cdot 1}$

Step 8.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i}{2 \cdot 1}$

Step 8.4.2

Multiply $2$ by $1$ .

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

Step 8.4.3

Simplify $\frac{- 2 \pm 2 i}{2}$ .

$x = - 1 \pm i$

Step 8.5

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

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Step 8.5.1

Simplify the numerator.

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Step 8.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 8.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 8.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 8.5.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 8.5.1.3

Subtract $8$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 8.5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 8.5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 8.5.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 8.5.1.7

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 8.5.1.8

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

$x = \frac{- 2 \pm i \cdot 2}{2 \cdot 1}$

Step 8.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i}{2 \cdot 1}$

Step 8.5.2

Multiply $2$ by $1$ .

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

Step 8.5.3

Simplify $\frac{- 2 \pm 2 i}{2}$ .

$x = - 1 \pm i$

Step 8.5.4

Change the $\pm$ to $+$ .

$x = - 1 + i$

Step 8.6

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

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Step 8.6.1

Simplify the numerator.

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Step 8.6.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 8.6.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 8.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 8.6.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 8.6.1.3

Subtract $8$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 8.6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 8.6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 8.6.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 8.6.1.7

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 8.6.1.8

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

$x = \frac{- 2 \pm i \cdot 2}{2 \cdot 1}$

Step 8.6.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i}{2 \cdot 1}$

Step 8.6.2

Multiply $2$ by $1$ .

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

Step 8.6.3

Simplify $\frac{- 2 \pm 2 i}{2}$ .

$x = - 1 \pm i$

Step 8.6.4

Change the $\pm$ to $-$ .

$x = - 1 - i$

Step 8.7

Identify the leading coefficient.

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Step 8.7.1

The leading term in a polynomial is the term with the highest degree.

$x^{2}$

Step 8.7.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$1$

Step 8.8

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $x^{2} + 2 x + 2$ is always greater than $0$ .

All real numbers

Step 9

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 10