Solve -3w^2+33w-54

$$- 3 w ^ { 2 } + 33 w - 54$$

Factor

$$3\left(2-w\right)\left(w-9\right)$$

Solution Steps

Factor out 3.

$$3\left(-w^{2}+11w-18\right)$$

Consider -w^{2}+11w-18. Factor the expression by grouping. First, the expression needs to be rewritten as -w^{2}+aw+bw-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 -w^{2}+11w-18 as \left(-w^{2}+9w\right)+\left(2w-18\right).

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

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

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

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

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

Rewrite the complete factored expression.

$$3\left(w-9\right)\left(-w+2\right)$$

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.

$$-3w^{2}+33w-54=0$$

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.

$$w=\frac{-33±\sqrt{33^{2}-4\left(-3\right)\left(-54\right)}}{2\left(-3\right)}$$

Square 33.

$$w=\frac{-33±\sqrt{1089-4\left(-3\right)\left(-54\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$w=\frac{-33±\sqrt{1089+12\left(-54\right)}}{2\left(-3\right)}$$

Multiply 12 times -54.

$$w=\frac{-33±\sqrt{1089-648}}{2\left(-3\right)}$$

Add 1089 to -648.

$$w=\frac{-33±\sqrt{441}}{2\left(-3\right)}$$

Take the square root of 441.

$$w=\frac{-33±21}{2\left(-3\right)}$$

Multiply 2 times -3.

$$w=\frac{-33±21}{-6}$$

Now solve the equation w=\frac{-33±21}{-6} when ± is plus. Add -33 to 21.

$$w=-\frac{12}{-6}$$

Divide -12 by -6.

$$w=2$$

Now solve the equation w=\frac{-33±21}{-6} when ± is minus. Subtract 21 from -33.

$$w=-\frac{54}{-6}$$

Divide -54 by -6.

$$w=9$$

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}.

$$-3w^{2}+33w-54=-3\left(w-2\right)\left(w-9\right)$$

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-gzdabgg4ehffg0hf.b01.azurefd.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

$$3\left(2-w\right)\left(w-9\right)$$