Solve B(x)=-x^2+11x-18

$$B ( x ) = - x ^ { 2 } + 11 x - 18$$

Factor

$$\left(2-x\right)\left(x-9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=11 ab=-\left(-18\right)=18$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 18.

$$1,18 2,9 3,6$$

Calculate the sum for each pair.

$$1+18=19 2+9=11 3+6=9$$

The solution is the pair that gives sum 11.

$$a=9 b=2$$

Rewrite -x^{2}+11x-18 as \left(-x^{2}+9x\right)+\left(2x-18\right).

$$\left(-x^{2}+9x\right)+\left(2x-18\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-9\right)+2\left(x-9\right)$$

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

$$\left(x-9\right)\left(-x+2\right)$$

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.

$$-x^{2}+11x-18=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-11±\sqrt{121+4\left(-18\right)}}{2\left(-1\right)}$$

Multiply 4 times -18.

$$x=\frac{-11±\sqrt{121-72}}{2\left(-1\right)}$$

Add 121 to -72.

$$x=\frac{-11±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of 49.

$$x=\frac{-11±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-11±7}{-2}$$

Now solve the equation x=\frac{-11±7}{-2} when ± is plus. Add -11 to 7.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-11±7}{-2} when ± is minus. Subtract 7 from -11.

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

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 2 for x_{1} and 9 for x_{2}.

$$-x^{2}+11x-18=-\left(x-2\right)\left(x-9\right)$$

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.

$$x ^ 2 -11x +18 = 0$$

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

$$r + s = 11 rs = 18$$

Two numbers r and s sum up to 11 exactly when the average of the two numbers is \frac{1}{2}*11 = \frac{11}{2}. 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>

$$r = \frac{11}{2} - u s = \frac{11}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 18

$$(\frac{11}{2} - u) (\frac{11}{2} + u) = 18$$

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

$$\frac{121}{4} - u^2 = 18$$

Simplify the expression by subtracting \frac{121}{4} on both sides

$$-u^2 = 18-\frac{121}{4} = -\frac{49}{4}$$

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

$$u^2 = \frac{49}{4} u = \pm\sqrt{\frac{49}{4}} = \pm \frac{7}{2}$$

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

$$r =\frac{11}{2} - \frac{7}{2} = 2 s = \frac{11}{2} + \frac{7}{2} = 9$$

Evaluate

$$\left(2-x\right)\left(x-9\right)$$