Solve -15*x^2+165x-270

$$-15 \times { x }^{ 2 } +165x-270$$

Factor

$$15\left(2-x\right)\left(x-9\right)$$

Solution Steps

Factor out 15.

$$15\left(-x^{2}+11x-18\right)$$

Consider -x^{2}+11x-18. 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.

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

Rewrite the complete factored expression.

$$15\left(x-9\right)\left(-x+2\right)$$

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.

$$-15x^{2}+165x-270=0$$

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.

$$x=\frac{-165±\sqrt{165^{2}-4\left(-15\right)\left(-270\right)}}{2\left(-15\right)}$$

Square 165.

$$x=\frac{-165±\sqrt{27225-4\left(-15\right)\left(-270\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-165±\sqrt{27225+60\left(-270\right)}}{2\left(-15\right)}$$

Multiply 60 times -270.

$$x=\frac{-165±\sqrt{27225-16200}}{2\left(-15\right)}$$

Add 27225 to -16200.

$$x=\frac{-165±\sqrt{11025}}{2\left(-15\right)}$$

Take the square root of 11025.

$$x=\frac{-165±105}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-165±105}{-30}$$

Now solve the equation x=\frac{-165±105}{-30} when ± is plus. Add -165 to 105.

$$x=-\frac{60}{-30}$$

Divide -60 by -30.

$$x=2$$

Now solve the equation x=\frac{-165±105}{-30} when ± is minus. Subtract 105 from -165.

$$x=-\frac{270}{-30}$$

Divide -270 by -30.

$$x=9$$

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}.

$$-15x^{2}+165x-270=-15\left(x-2\right)\left(x-9\right)$$

Evaluate

$$15\left(2-x\right)\left(x-9\right)$$