Solve f(x)=-x^2+6x+27

$$f ( x ) = - x ^ { 2 } + 6 x + 27$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+27. To find a and b, set up a system to be solved.

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

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.

$$a=9 b=-3$$

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

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

Factor out -x in the first and -3 in the second group.

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

Factor out common term x-9 by using distributive property.

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

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

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

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 27}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\times 27}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\times 27}}{2\left(-1\right)}$$

Multiply 4 times 27.

$$x=\frac{-6±\sqrt{36+108}}{2\left(-1\right)}$$

Add 36 to 108.

$$x=\frac{-6±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$x=\frac{-6±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±12}{-2}$$

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.

$$x=9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and 9 for x_{2}.

$$-x^{2}+6x+27=-\left(x-\left(-3\right)\right)\left(x-9\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-x^{2}+6x+27=-\left(x+3\right)\left(x-9\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -6x -27 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 6 rs = -27$$

Two numbers r and s sum up to 6 exactly when the average of the two numbers is \frac{1}{2}*6 = 3. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 3 - u s = 3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -27

$$(3 - u) (3 + u) = -27$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$9 - u^2 = -27$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -27-9 = -36$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 36 u = \pm\sqrt{36} = \pm 6$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =3 - 6 = -3 s = 3 + 6 = 9$$

Evaluate

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