Find the x and y Intercepts x^2+y^2-6x-4y-23=0

$x^{2} + y^{2} - 6 x - 4 y - 23 = 0$

Step 1

Find the x-intercepts.

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

$x^{2} + {(0)}^{2} - 6 x - 4 \cdot 0 - 23 = 0$

Solve the equation.

Simplify $x^{2} + {(0)}^{2} - 6 x - 4 \cdot 0 - 23$ .

Simplify each term.

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

$x^{2} + 0 - 6 x - 4 \cdot 0 - 23 = 0$

Multiply $- 4$ by $0$ .

$x^{2} + 0 - 6 x + 0 - 23 = 0$

Combine the opposite terms in $x^{2} + 0 - 6 x + 0 - 23$ .

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

$x^{2} - 6 x + 0 - 23 = 0$

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

$x^{2} - 6 x - 23 = 0$

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 $c = - 23$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 23)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 23$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4$ by $- 23$ .

$x = \frac{6 \pm \sqrt{36 + 92}}{2 \cdot 1}$

Add $36$ and $92$ .

$x = \frac{6 \pm \sqrt{128}}{2 \cdot 1}$

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$x = \frac{6 \pm \sqrt{64 (2)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 8 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 8 \sqrt{2}}{2}$

Simplify $\frac{6 \pm 8 \sqrt{2}}{2}$ .

$x = 3 \pm 4 \sqrt{2}$

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

Simplify the numerator.

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 23$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4$ by $- 23$ .

$x = \frac{6 \pm \sqrt{36 + 92}}{2 \cdot 1}$

Add $36$ and $92$ .

$x = \frac{6 \pm \sqrt{128}}{2 \cdot 1}$

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$x = \frac{6 \pm \sqrt{64 (2)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 8 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 8 \sqrt{2}}{2}$

Simplify $\frac{6 \pm 8 \sqrt{2}}{2}$ .

$x = 3 \pm 4 \sqrt{2}$

Change the $\pm$ to $+$ .

$x = 3 + 4 \sqrt{2}$

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

Simplify the numerator.

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 23$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4$ by $- 23$ .

$x = \frac{6 \pm \sqrt{36 + 92}}{2 \cdot 1}$

Add $36$ and $92$ .

$x = \frac{6 \pm \sqrt{128}}{2 \cdot 1}$

Rewrite $128$ as $8^{2} \cdot 2$ .

Factor $64$ out of $128$ .

$x = \frac{6 \pm \sqrt{64 (2)}}{2 \cdot 1}$

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

$x = \frac{6 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 8 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 8 \sqrt{2}}{2}$

Simplify $\frac{6 \pm 8 \sqrt{2}}{2}$ .

$x = 3 \pm 4 \sqrt{2}$

Change the $\pm$ to $-$ .

$x = 3 - 4 \sqrt{2}$

The final answer is the combination of both solutions.

$x = 3 + 4 \sqrt{2}, 3 - 4 \sqrt{2}$

x-intercept(s) in point form.

x-intercept(s): $(3 + 4 \sqrt{2}, 0), (3 - 4 \sqrt{2}, 0)$

Step 2

Find the y-intercepts.

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

${(0)}^{2} + y^{2} - 6 \cdot 0 - 4 y - 23 = 0$

Solve the equation.

Simplify ${(0)}^{2} + y^{2} - 6 \cdot 0 - 4 y - 23$ .

Simplify each term.

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

$0 + y^{2} - 6 \cdot 0 - 4 y - 23 = 0$

Multiply $- 6$ by $0$ .

$0 + y^{2} + 0 - 4 y - 23 = 0$

Combine the opposite terms in $0 + y^{2} + 0 - 4 y - 23$ .

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

$y^{2} + 0 - 4 y - 23 = 0$

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

$y^{2} - 4 y - 23 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 23$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 23)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 23$ .

Multiply $- 4$ by $1$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4$ by $- 23$ .

$y = \frac{4 \pm \sqrt{16 + 92}}{2 \cdot 1}$

Add $16$ and $92$ .

$y = \frac{4 \pm \sqrt{108}}{2 \cdot 1}$

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

Factor $36$ out of $108$ .

$y = \frac{4 \pm \sqrt{36 (3)}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{4 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{4 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{4 \pm 6 \sqrt{3}}{2}$ .

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

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 23$ .

Multiply $- 4$ by $1$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4$ by $- 23$ .

$y = \frac{4 \pm \sqrt{16 + 92}}{2 \cdot 1}$

Add $16$ and $92$ .

$y = \frac{4 \pm \sqrt{108}}{2 \cdot 1}$

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

Factor $36$ out of $108$ .

$y = \frac{4 \pm \sqrt{36 (3)}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{4 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{4 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{4 \pm 6 \sqrt{3}}{2}$ .

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

Change the $\pm$ to $+$ .

$y = 2 + 3 \sqrt{3}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 23$ .

Multiply $- 4$ by $1$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot - 23}}{2 \cdot 1}$

Multiply $- 4$ by $- 23$ .

$y = \frac{4 \pm \sqrt{16 + 92}}{2 \cdot 1}$

Add $16$ and $92$ .

$y = \frac{4 \pm \sqrt{108}}{2 \cdot 1}$

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

Factor $36$ out of $108$ .

$y = \frac{4 \pm \sqrt{36 (3)}}{2 \cdot 1}$

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

$y = \frac{4 \pm \sqrt{6^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{4 \pm 6 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{4 \pm 6 \sqrt{3}}{2}$

Simplify $\frac{4 \pm 6 \sqrt{3}}{2}$ .

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

Change the $\pm$ to $-$ .

$y = 2 - 3 \sqrt{3}$

The final answer is the combination of both solutions.

$y = 2 + 3 \sqrt{3}, 2 - 3 \sqrt{3}$

y-intercept(s) in point form.

y-intercept(s): $(0, 2 + 3 \sqrt{3}), (0, 2 - 3 \sqrt{3})$

Step 3

List the intersections.

x-intercept(s): $(3 + 4 \sqrt{2}, 0), (3 - 4 \sqrt{2}, 0)$

y-intercept(s): $(0, 2 + 3 \sqrt{3}), (0, 2 - 3 \sqrt{3})$

Step 4

Find the x and y Intercepts x^2+y^2-6x-4y-23=0

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