Solve -5u^2+45u-90

$$- 5 u ^ { 2 } + 45 u - 90$$

Factor

$$5\left(3-u\right)\left(u-6\right)$$

Solution Steps

Factor out 5.

$$5\left(-u^{2}+9u-18\right)$$

Consider -u^{2}+9u-18. Factor the expression by grouping. First, the expression needs to be rewritten as -u^{2}+au+bu-18. To find a and b, set up a system to be solved.

$$a+b=9 ab=-\left(-18\right)=18$$

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 9.

$$a=6 b=3$$

Rewrite -u^{2}+9u-18 as \left(-u^{2}+6u\right)+\left(3u-18\right).

$$\left(-u^{2}+6u\right)+\left(3u-18\right)$$

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

$$-u\left(u-6\right)+3\left(u-6\right)$$

Factor out common term u-6 by using distributive property.

$$\left(u-6\right)\left(-u+3\right)$$

Rewrite the complete factored expression.

$$5\left(u-6\right)\left(-u+3\right)$$

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.

$$-5u^{2}+45u-90=0$$

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.

$$u=\frac{-45±\sqrt{45^{2}-4\left(-5\right)\left(-90\right)}}{2\left(-5\right)}$$

Square 45.

$$u=\frac{-45±\sqrt{2025-4\left(-5\right)\left(-90\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$u=\frac{-45±\sqrt{2025+20\left(-90\right)}}{2\left(-5\right)}$$

Multiply 20 times -90.

$$u=\frac{-45±\sqrt{2025-1800}}{2\left(-5\right)}$$

Add 2025 to -1800.

$$u=\frac{-45±\sqrt{225}}{2\left(-5\right)}$$

Take the square root of 225.

$$u=\frac{-45±15}{2\left(-5\right)}$$

Multiply 2 times -5.

$$u=\frac{-45±15}{-10}$$

Now solve the equation u=\frac{-45±15}{-10} when ± is plus. Add -45 to 15.

$$u=-\frac{30}{-10}$$

Divide -30 by -10.

$$u=3$$

Now solve the equation u=\frac{-45±15}{-10} when ± is minus. Subtract 15 from -45.

$$u=-\frac{60}{-10}$$

Divide -60 by -10.

$$u=6$$

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 6 for x_{2}.

$$-5u^{2}+45u-90=-5\left(u-3\right)\left(u-6\right)$$

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.

$$x ^ 2 -9x +18 = 0$$

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

$$r + s = 9 rs = 18$$

Two numbers r and s sum up to 9 exactly when the average of the two numbers is \frac{1}{2}*9 = \frac{9}{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>

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

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

$$(\frac{9}{2} - u) (\frac{9}{2} + u) = 18$$

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

$$\frac{81}{4} - u^2 = 18$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = 18-\frac{81}{4} = -\frac{9}{4}$$

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

$$u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{2}$$

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

$$r =\frac{9}{2} - \frac{3}{2} = 3 s = \frac{9}{2} + \frac{3}{2} = 6$$

Evaluate

$$5\left(3-u\right)\left(u-6\right)$$