Solve 6x-27=-x^2

$$6x-27=- { x }^{ 2 }$$

Solve for x

$$x=-9$$

$$x=3$$

Steps Using Factoring

Add x^{2} to both sides.

$$6x-27+x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+6x-27=0$$

To solve the equation, factor x^{2}+6x-27 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=6 ab=-27$$

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. List all such integer pairs that give product -27.

$$-1,27 -3,9$$

Calculate the sum for each pair.

$$-1+27=26 -3+9=6$$

The solution is the pair that gives sum 6.

$$a=-3 b=9$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-3\right)\left(x+9\right)$$

To find equation solutions, solve x-3=0 and x+9=0.

$$x=3 x=-9$$

Steps Using Factoring By Grouping

Add x^{2} to both sides.

$$6x-27+x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+6x-27=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-27. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\left(-27\right)=-27$$

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. List all such integer pairs that give product -27.

$$-1,27 -3,9$$

Calculate the sum for each pair.

$$-1+27=26 -3+9=6$$

The solution is the pair that gives sum 6.

$$a=-3 b=9$$

Rewrite x^{2}+6x-27 as \left(x^{2}-3x\right)+\left(9x-27\right).

$$\left(x^{2}-3x\right)+\left(9x-27\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-3\right)+9\left(x-3\right)$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(x+9\right)$$

To find equation solutions, solve x-3=0 and x+9=0.

$$x=3 x=-9$$

Steps Using the Quadratic Formula

Add x^{2} to both sides.

$$6x-27+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+6x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-27\right)}}{2}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-27\right)}}{2}$$

Multiply -4 times -27.

$$x=\frac{-6±\sqrt{36+108}}{2}$$

Add 36 to 108.

$$x=\frac{-6±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-6±12}{2}$$

Now solve the equation x=\frac{-6±12}{2} when ± is plus. Add -6 to 12.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

Now solve the equation x=\frac{-6±12}{2} when ± is minus. Subtract 12 from -6.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=3 x=-9$$

Steps for Completing the Square

Add x^{2} to both sides.

$$6x-27+x^{2}=0$$

Add 27 to both sides. Anything plus zero gives itself.

$$6x+x^{2}=27$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+6x=27$$

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.

$$x^{2}+6x+3^{2}=27+3^{2}$$

Square 3.

$$x^{2}+6x+9=27+9$$

Add 27 to 9.

$$x^{2}+6x+9=36$$

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}.

$$\left(x+3\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+3=6 x+3=-6$$

Subtract 3 from both sides of the equation.

$$x=3 x=-9$$