Linear Algebra Examples

$x^{2} + y^{2} - 2 x + 4 x - 6361 = 0$

Step 1

Add $- 2 x$ and $4 x$ .

$x^{2} + y^{2} + 2 x - 6361 = 0$

Step 2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 2.1

Subtract $x^{2}$ from both sides of the equation.

$y^{2} + 2 x - 6361 = - x^{2}$

Step 2.2

Subtract $2 x$ from both sides of the equation.

$y^{2} - 6361 = - x^{2} - 2 x$

Step 2.3

Add $6361$ to both sides of the equation.

$y^{2} = - x^{2} - 2 x + 6361$

Step 3

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

$y = \pm \sqrt{- x^{2} - 2 x + 6361}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{- x^{2} - 2 x + 6361}$

Step 4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{- x^{2} - 2 x + 6361}$

Step 4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{- x^{2} - 2 x + 6361}$

$y = - \sqrt{- x^{2} - 2 x + 6361}$

$y = \sqrt{- x^{2} - 2 x + 6361}$

$y = - \sqrt{- x^{2} - 2 x + 6361}$

Step 5

Set the radicand in $\sqrt{- x^{2} - 2 x + 6361}$ greater than or equal to $0$ to find where the expression is defined.

$- x^{2} - 2 x + 6361 \geq 0$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Convert the inequality to an equation.

$- x^{2} - 2 x + 6361 = 0$

Step 6.2

Use the quadratic formula to find the solutions.

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

Step 6.3

Substitute the values $a = - 1$ , $b = - 2$ , and $c = 6361$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (- 1 \cdot 6361)}}{2 \cdot - 1}$

Step 6.4

Simplify.

Tap for more steps...

Step 6.4.1

Simplify the numerator.

Tap for more steps...

Step 6.4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot 6361}}{2 \cdot - 1}$

Step 6.4.1.2

Multiply $- 4 \cdot - 1 \cdot 6361$ .

Tap for more steps...

Step 6.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4 \cdot 6361}}{2 \cdot - 1}$

Step 6.4.1.2.2

Multiply $4$ by $6361$ .

$x = \frac{2 \pm \sqrt{4 + 25444}}{2 \cdot - 1}$

Step 6.4.1.3

Add $4$ and $25444$ .

$x = \frac{2 \pm \sqrt{25448}}{2 \cdot - 1}$

Step 6.4.1.4

Rewrite $25448$ as $2^{2} \cdot 6362$ .

Tap for more steps...

Step 6.4.1.4.1

Factor $4$ out of $25448$ .

$x = \frac{2 \pm \sqrt{4 (6362)}}{2 \cdot - 1}$

Step 6.4.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 6362}}{2 \cdot - 1}$

Step 6.4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{6362}}{2 \cdot - 1}$

Step 6.4.2

Multiply $2$ by $- 1$ .

$x = \frac{2 \pm 2 \sqrt{6362}}{- 2}$

Step 6.4.3

Simplify $\frac{2 \pm 2 \sqrt{6362}}{- 2}$ .

$x = \frac{1 \pm \sqrt{6362}}{- 1}$

Step 6.4.4

Move the negative one from the denominator of $\frac{1 \pm \sqrt{6362}}{- 1}$ .

$x = - 1 \cdot (1 \pm \sqrt{6362})$

Step 6.4.5

Rewrite $- 1 \cdot (1 \pm \sqrt{6362})$ as $- (1 \pm \sqrt{6362})$ .

$x = - (1 \pm \sqrt{6362})$

Step 6.5

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

Tap for more steps...

Step 6.5.1

Simplify the numerator.

Tap for more steps...

Step 6.5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot 6361}}{2 \cdot - 1}$

Step 6.5.1.2

Multiply $- 4 \cdot - 1 \cdot 6361$ .

Tap for more steps...

Step 6.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4 \cdot 6361}}{2 \cdot - 1}$

Step 6.5.1.2.2

Multiply $4$ by $6361$ .

$x = \frac{2 \pm \sqrt{4 + 25444}}{2 \cdot - 1}$

Step 6.5.1.3

Add $4$ and $25444$ .

$x = \frac{2 \pm \sqrt{25448}}{2 \cdot - 1}$

Step 6.5.1.4

Rewrite $25448$ as $2^{2} \cdot 6362$ .

Tap for more steps...

Step 6.5.1.4.1

Factor $4$ out of $25448$ .

$x = \frac{2 \pm \sqrt{4 (6362)}}{2 \cdot - 1}$

Step 6.5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 6362}}{2 \cdot - 1}$

Step 6.5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{6362}}{2 \cdot - 1}$

Step 6.5.2

Multiply $2$ by $- 1$ .

$x = \frac{2 \pm 2 \sqrt{6362}}{- 2}$

Step 6.5.3

Simplify $\frac{2 \pm 2 \sqrt{6362}}{- 2}$ .

$x = \frac{1 \pm \sqrt{6362}}{- 1}$

Step 6.5.4

Move the negative one from the denominator of $\frac{1 \pm \sqrt{6362}}{- 1}$ .

$x = - 1 \cdot (1 \pm \sqrt{6362})$

Step 6.5.5

Rewrite $- 1 \cdot (1 \pm \sqrt{6362})$ as $- (1 \pm \sqrt{6362})$ .

$x = - (1 \pm \sqrt{6362})$

Step 6.5.6

Change the $\pm$ to $+$ .

$x = - (1 + \sqrt{6362})$

Step 6.5.7

Apply the distributive property.

$x = - 1 \cdot 1 - \sqrt{6362}$

Step 6.5.8

Multiply $- 1$ by $1$ .

$x = - 1 - \sqrt{6362}$

Step 6.6

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

Tap for more steps...

Step 6.6.1

Simplify the numerator.

Tap for more steps...

Step 6.6.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot 6361}}{2 \cdot - 1}$

Step 6.6.1.2

Multiply $- 4 \cdot - 1 \cdot 6361$ .

Tap for more steps...

Step 6.6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4 \cdot 6361}}{2 \cdot - 1}$

Step 6.6.1.2.2

Multiply $4$ by $6361$ .

$x = \frac{2 \pm \sqrt{4 + 25444}}{2 \cdot - 1}$

Step 6.6.1.3

Add $4$ and $25444$ .

$x = \frac{2 \pm \sqrt{25448}}{2 \cdot - 1}$

Step 6.6.1.4

Rewrite $25448$ as $2^{2} \cdot 6362$ .

Tap for more steps...

Step 6.6.1.4.1

Factor $4$ out of $25448$ .

$x = \frac{2 \pm \sqrt{4 (6362)}}{2 \cdot - 1}$

Step 6.6.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 6362}}{2 \cdot - 1}$

Step 6.6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{6362}}{2 \cdot - 1}$

Step 6.6.2

Multiply $2$ by $- 1$ .

$x = \frac{2 \pm 2 \sqrt{6362}}{- 2}$

Step 6.6.3

Simplify $\frac{2 \pm 2 \sqrt{6362}}{- 2}$ .

$x = \frac{1 \pm \sqrt{6362}}{- 1}$

Step 6.6.4

Move the negative one from the denominator of $\frac{1 \pm \sqrt{6362}}{- 1}$ .

$x = - 1 \cdot (1 \pm \sqrt{6362})$

Step 6.6.5

Rewrite $- 1 \cdot (1 \pm \sqrt{6362})$ as $- (1 \pm \sqrt{6362})$ .

$x = - (1 \pm \sqrt{6362})$

Step 6.6.6

Change the $\pm$ to $-$ .

$x = - (1 - \sqrt{6362})$

Step 6.6.7

Apply the distributive property.

$x = - 1 \cdot 1 + \sqrt{6362}$

Step 6.6.8

Multiply $- 1$ by $1$ .

$x = - 1 + \sqrt{6362}$

Step 6.6.9

Multiply $- - \sqrt{6362}$ .

Tap for more steps...

Step 6.6.9.1

Multiply $- 1$ by $- 1$ .

$x = - 1 + 1 \sqrt{6362}$

Step 6.6.9.2

Multiply $\sqrt{6362}$ by $1$ .

$x = - 1 + \sqrt{6362}$

Step 6.7

Consolidate the solutions.

$x = - 1 - \sqrt{6362}, - 1 + \sqrt{6362}$

Step 6.8

Use each root to create test intervals.

$x < - 1 - \sqrt{6362}$

$- 1 - \sqrt{6362} < x < - 1 + \sqrt{6362}$

$x > - 1 + \sqrt{6362}$

Step 6.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 6.9.1

Test a value on the interval $x < - 1 - \sqrt{6362}$ to see if it makes the inequality true.

Tap for more steps...

Step 6.9.1.1

Choose a value on the interval $x < - 1 - \sqrt{6362}$ and see if this value makes the original inequality true.

$x = - 83$

Step 6.9.1.2

Replace $x$ with $- 83$ in the original inequality.

$- {(- 83)}^{2} - 2 \cdot - 83 + 6361 \geq 0$

Step 6.9.1.3

The left side $- 362$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.9.2

Test a value on the interval $- 1 - \sqrt{6362} < x < - 1 + \sqrt{6362}$ to see if it makes the inequality true.

Tap for more steps...

Step 6.9.2.1

Choose a value on the interval $- 1 - \sqrt{6362} < x < - 1 + \sqrt{6362}$ and see if this value makes the original inequality true.

$x = 0$

Step 6.9.2.2

Replace $x$ with $0$ in the original inequality.

$- {(0)}^{2} - 2 \cdot 0 + 6361 \geq 0$

Step 6.9.2.3

The left side $6361$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.9.3

Test a value on the interval $x > - 1 + \sqrt{6362}$ to see if it makes the inequality true.

Tap for more steps...

Step 6.9.3.1

Choose a value on the interval $x > - 1 + \sqrt{6362}$ and see if this value makes the original inequality true.

$x = 81$

Step 6.9.3.2

Replace $x$ with $81$ in the original inequality.

$- {(81)}^{2} - 2 \cdot 81 + 6361 \geq 0$

Step 6.9.3.3

The left side $- 362$ is less than the right side $0$ , which means that the given statement is false.

False

Step 6.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1 - \sqrt{6362}$ False

$- 1 - \sqrt{6362} < x < - 1 + \sqrt{6362}$ True

$x > - 1 + \sqrt{6362}$ False

$x < - 1 - \sqrt{6362}$ False

$- 1 - \sqrt{6362} < x < - 1 + \sqrt{6362}$ True

$x > - 1 + \sqrt{6362}$ False

Step 6.10

The solution consists of all of the true intervals.

$- 1 - \sqrt{6362} \leq x \leq - 1 + \sqrt{6362}$

Step 7

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

Interval Notation:

$[- 1 - \sqrt{6362}, - 1 + \sqrt{6362}]$

Set-Builder Notation:

${x | - 1 - \sqrt{6362} \leq x \leq - 1 + \sqrt{6362}}$

Step 8