Find the x and y Intercepts y=-x^2+6x-7

$y = - x^{2} + 6 x - 7$

Step 1

Find the x-intercepts.

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

$0 = - x^{2} + 6 x - 7$

Solve the equation.

Rewrite the equation as $- x^{2} + 6 x - 7 = 0$ .

$- x^{2} + 6 x - 7 = 0$

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 $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 1 \cdot - 7)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 7}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 7$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 7}}{2 \cdot - 1}$

Multiply $4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 - 28}}{2 \cdot - 1}$

Subtract $28$ from $36$ .

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

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

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

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

Simplify the numerator.

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 7}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 7$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 7}}{2 \cdot - 1}$

Multiply $4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 - 28}}{2 \cdot - 1}$

Subtract $28$ from $36$ .

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

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

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

Change the $\pm$ to $+$ .

$x = 3 + \sqrt{2}$

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

Simplify the numerator.

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 7}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 7$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 7}}{2 \cdot - 1}$

Multiply $4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 - 28}}{2 \cdot - 1}$

Subtract $28$ from $36$ .

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

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

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

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

Change the $\pm$ to $-$ .

$x = 3 - \sqrt{2}$

The final answer is the combination of both solutions.

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

x-intercept(s) in point form.

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

Step 2

Find the y-intercepts.

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

$y = - {(0)}^{2} + 6 (0) - 7$

Solve the equation.

Remove parentheses.

$y = - 0^{2} + 6 (0) - 7$

Remove parentheses.

$y = - {(0)}^{2} + 6 (0) - 7$

Simplify $- {(0)}^{2} + 6 (0) - 7$ .

Simplify each term.

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

$y = - 0 + 6 (0) - 7$

Multiply $- 1$ by $0$ .

$y = 0 + 6 (0) - 7$

Multiply $6$ by $0$ .

$y = 0 + 0 - 7$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 7$

Subtract $7$ from $0$ .

$y = - 7$

y-intercept(s) in point form.

y-intercept(s): $(0, - 7)$

Step 3

List the intersections.

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

y-intercept(s): $(0, - 7)$

Step 4

Find the x and y Intercepts y=-x^2+6x-7

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