Solve 0=-x^2+6x+7

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

Solve for x

$$x=-1$$

$$x=7$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=6 ab=-7=-7$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=7 b=-1$$

Rewrite -x^{2}+6x+7 as \left(-x^{2}+7x\right)+\left(-x+7\right).

$$\left(-x^{2}+7x\right)+\left(-x+7\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(x-7\right)-\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x-1\right)$$

To find equation solutions, solve x-7=0 and -x-1=0.

$$x=7 x=-1$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\times 7}}{2\left(-1\right)}$$

Multiply 4 times 7.

$$x=\frac{-6±\sqrt{36+28}}{2\left(-1\right)}$$

Add 36 to 28.

$$x=\frac{-6±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-6±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±8}{-2}$$

Now solve the equation x=\frac{-6±8}{-2} when ± is plus. Add -6 to 8.

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

Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{-6±8}{-2} when ± is minus. Subtract 8 from -6.

$$x=-\frac{14}{-2}$$

Divide -14 by -2.

$$x=7$$

The equation is now solved.

$$x=-1 x=7$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

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

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -1.

$$\frac{-x^{2}+6x}{-1}=-\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{6}{-1}x=-\frac{7}{-1}$$

Divide 6 by -1.

$$x^{2}-6x=-\frac{7}{-1}$$

Divide -7 by -1.

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

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=7+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=7+9$$

Add 7 to 9.

$$x^{2}-6x+9=16$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-3=4 x-3=-4$$

Add 3 to both sides of the equation.

$$x=7 x=-1$$