Solve u^2-9u+27=(2u-9)(u+2)

$$u ^ { 2 } - 9 u + 27 = ( 2 u - 9 ) ( u + 2 )$$

Solve for u

$$u=-9$$

$$u=5$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 2u-9 by u+2 and combine like terms.

$$u^{2}-9u+27=2u^{2}-5u-18$$

Subtract 2u^{2} from both sides.

$$u^{2}-9u+27-2u^{2}=-5u-18$$

Combine u^{2} and -2u^{2} to get -u^{2}.

$$-u^{2}-9u+27=-5u-18$$

Add 5u to both sides.

$$-u^{2}-9u+27+5u=-18$$

Combine -9u and 5u to get -4u.

$$-u^{2}-4u+27=-18$$

Add 18 to both sides.

$$-u^{2}-4u+27+18=0$$

Add 27 and 18 to get 45.

$$-u^{2}-4u+45=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -u^{2}+au+bu+45. To find a and b, set up a system to be solved.

$$a+b=-4 ab=-45=-45$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -45.

$$1,-45 3,-15 5,-9$$

Calculate the sum for each pair.

$$1-45=-44 3-15=-12 5-9=-4$$

The solution is the pair that gives sum -4.

$$a=5 b=-9$$

Rewrite -u^{2}-4u+45 as \left(-u^{2}+5u\right)+\left(-9u+45\right).

$$\left(-u^{2}+5u\right)+\left(-9u+45\right)$$

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

$$u\left(-u+5\right)+9\left(-u+5\right)$$

Factor out common term -u+5 by using distributive property.

$$\left(-u+5\right)\left(u+9\right)$$

To find equation solutions, solve -u+5=0 and u+9=0.

$$u=5 u=-9$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2u-9 by u+2 and combine like terms.

$$u^{2}-9u+27=2u^{2}-5u-18$$

Subtract 2u^{2} from both sides.

$$u^{2}-9u+27-2u^{2}=-5u-18$$

Combine u^{2} and -2u^{2} to get -u^{2}.

$$-u^{2}-9u+27=-5u-18$$

Add 5u to both sides.

$$-u^{2}-9u+27+5u=-18$$

Combine -9u and 5u to get -4u.

$$-u^{2}-4u+27=-18$$

Add 18 to both sides.

$$-u^{2}-4u+27+18=0$$

Add 27 and 18 to get 45.

$$-u^{2}-4u+45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -4 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 45}}{2\left(-1\right)}$$

Square -4.

$$u=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 45}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$u=\frac{-\left(-4\right)±\sqrt{16+4\times 45}}{2\left(-1\right)}$$

Multiply 4 times 45.

$$u=\frac{-\left(-4\right)±\sqrt{16+180}}{2\left(-1\right)}$$

Add 16 to 180.

$$u=\frac{-\left(-4\right)±\sqrt{196}}{2\left(-1\right)}$$

Take the square root of 196.

$$u=\frac{-\left(-4\right)±14}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$u=\frac{4±14}{2\left(-1\right)}$$

Multiply 2 times -1.

$$u=\frac{4±14}{-2}$$

Now solve the equation u=\frac{4±14}{-2} when ± is plus. Add 4 to 14.

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

Divide 18 by -2.

$$u=-9$$

Now solve the equation u=\frac{4±14}{-2} when ± is minus. Subtract 14 from 4.

$$u=-\frac{10}{-2}$$

Divide -10 by -2.

$$u=5$$

The equation is now solved.

$$u=-9 u=5$$

Steps for Completing the Square

Use the distributive property to multiply 2u-9 by u+2 and combine like terms.

$$u^{2}-9u+27=2u^{2}-5u-18$$

Subtract 2u^{2} from both sides.

$$u^{2}-9u+27-2u^{2}=-5u-18$$

Combine u^{2} and -2u^{2} to get -u^{2}.

$$-u^{2}-9u+27=-5u-18$$

Add 5u to both sides.

$$-u^{2}-9u+27+5u=-18$$

Combine -9u and 5u to get -4u.

$$-u^{2}-4u+27=-18$$

Subtract 27 from both sides.

$$-u^{2}-4u=-18-27$$

Subtract 27 from -18 to get -45.

$$-u^{2}-4u=-45$$

Divide both sides by -1.

$$\frac{-u^{2}-4u}{-1}=-\frac{45}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$u^{2}+\left(-\frac{4}{-1}\right)u=-\frac{45}{-1}$$

Divide -4 by -1.

$$u^{2}+4u=-\frac{45}{-1}$$

Divide -45 by -1.

$$u^{2}+4u=45$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}+4u+2^{2}=45+2^{2}$$

Square 2.

$$u^{2}+4u+4=45+4$$

Add 45 to 4.

$$u^{2}+4u+4=49$$

Factor u^{2}+4u+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(u+2\right)^{2}=49$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u+2\right)^{2}}=\sqrt{49}$$

Simplify.

$$u+2=7 u+2=-7$$

Subtract 2 from both sides of the equation.

$$u=5 u=-9$$