Find the x and y Intercepts x^2+y^2-6x-2y-26=0

$x^{2} + y^{2} - 6 x - 2 y - 26 = 0$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$x^{2} + {(0)}^{2} - 6 x - 2 \cdot 0 - 26 = 0$

Solve the equation.

Simplify $x^{2} + {(0)}^{2} - 6 x - 2 \cdot 0 - 26$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$x^{2} + 0 - 6 x - 2 \cdot 0 - 26 = 0$

Multiply $- 2$ by $0$ .

$x^{2} + 0 - 6 x + 0 - 26 = 0$

Combine the opposite terms in $x^{2} + 0 - 6 x + 0 - 26$ .

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

$x^{2} - 6 x + 0 - 26 = 0$

Add $x^{2} - 6 x$ and $0$ .

$x^{2} - 6 x - 26 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 6$ , and $c = - 26$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 26)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 26$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4$ by $- 26$ .

$x = \frac{6 \pm \sqrt{36 + 104}}{2 \cdot 1}$

Add $36$ and $104$ .

$x = \frac{6 \pm \sqrt{140}}{2 \cdot 1}$

Rewrite $140$ as $2^{2} \cdot 35$ .

Factor $4$ out of $140$ .

$x = \frac{6 \pm \sqrt{4 (35)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{35}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{35}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{35}}{2}$ .

$x = 3 \pm \sqrt{35}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 26$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4$ by $- 26$ .

$x = \frac{6 \pm \sqrt{36 + 104}}{2 \cdot 1}$

Add $36$ and $104$ .

$x = \frac{6 \pm \sqrt{140}}{2 \cdot 1}$

Rewrite $140$ as $2^{2} \cdot 35$ .

Factor $4$ out of $140$ .

$x = \frac{6 \pm \sqrt{4 (35)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{35}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{35}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{35}}{2}$ .

$x = 3 \pm \sqrt{35}$

Change the $\pm$ to $+$ .

$x = 3 + \sqrt{35}$

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

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 26$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4$ by $- 26$ .

$x = \frac{6 \pm \sqrt{36 + 104}}{2 \cdot 1}$

Add $36$ and $104$ .

$x = \frac{6 \pm \sqrt{140}}{2 \cdot 1}$

Rewrite $140$ as $2^{2} \cdot 35$ .

Factor $4$ out of $140$ .

$x = \frac{6 \pm \sqrt{4 (35)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 35}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{35}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{35}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{35}}{2}$ .

$x = 3 \pm \sqrt{35}$

Change the $\pm$ to $-$ .

$x = 3 - \sqrt{35}$

The final answer is the combination of both solutions.

$x = 3 + \sqrt{35}, 3 - \sqrt{35}$

x-intercept(s) in point form.

x-intercept(s): $(3 + \sqrt{35}, 0), (3 - \sqrt{35}, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

${(0)}^{2} + y^{2} - 6 \cdot 0 - 2 y - 26 = 0$

Solve the equation.

Simplify ${(0)}^{2} + y^{2} - 6 \cdot 0 - 2 y - 26$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$0 + y^{2} - 6 \cdot 0 - 2 y - 26 = 0$

Multiply $- 6$ by $0$ .

$0 + y^{2} + 0 - 2 y - 26 = 0$

Combine the opposite terms in $0 + y^{2} + 0 - 2 y - 26$ .

Add $0$ and $y^{2}$ .

$y^{2} + 0 - 2 y - 26 = 0$

Add $y^{2}$ and $0$ .

$y^{2} - 2 y - 26 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 26$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 26)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 26$ .

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4$ by $- 26$ .

$y = \frac{2 \pm \sqrt{4 + 104}}{2 \cdot 1}$

Add $4$ and $104$ .

$y = \frac{2 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$y = \frac{2 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$y = \frac{2 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{2 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{2 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{2 \pm 6 \sqrt{3}}{2}$ .

$y = 1 \pm 3 \sqrt{3}$

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

Simplify the numerator.

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 26$ .

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4$ by $- 26$ .

$y = \frac{2 \pm \sqrt{4 + 104}}{2 \cdot 1}$

Add $4$ and $104$ .

$y = \frac{2 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$y = \frac{2 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$y = \frac{2 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{2 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{2 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{2 \pm 6 \sqrt{3}}{2}$ .

$y = 1 \pm 3 \sqrt{3}$

Change the $\pm$ to $+$ .

$y = 1 + 3 \sqrt{3}$

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

Simplify the numerator.

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 26$ .

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot - 26}}{2 \cdot 1}$

Multiply $- 4$ by $- 26$ .

$y = \frac{2 \pm \sqrt{4 + 104}}{2 \cdot 1}$

Add $4$ and $104$ .

$y = \frac{2 \pm \sqrt{108}}{2 \cdot 1}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$y = \frac{2 \pm \sqrt{36 (3)}}{2 \cdot 1}$

Rewrite $36$ as $6^{2}$ .

$y = \frac{2 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{2 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{2 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{2 \pm 6 \sqrt{3}}{2}$ .

$y = 1 \pm 3 \sqrt{3}$

Change the $\pm$ to $-$ .

$y = 1 - 3 \sqrt{3}$

The final answer is the combination of both solutions.

$y = 1 + 3 \sqrt{3}, 1 - 3 \sqrt{3}$

y-intercept(s) in point form.

y-intercept(s): $(0, 1 + 3 \sqrt{3}), (0, 1 - 3 \sqrt{3})$

Step 3

List the intersections.

x-intercept(s): $(3 + \sqrt{35}, 0), (3 - \sqrt{35}, 0)$

y-intercept(s): $(0, 1 + 3 \sqrt{3}), (0, 1 - 3 \sqrt{3})$

Step 4

Find the x and y Intercepts x^2+y^2-6x-2y-26=0

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