Solve -5x^2-21x+54

$$-5 { x }^{ 2 } -21x+54$$

Factor

$$-\left(5x-9\right)\left(x+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -5x^{2}+ax+bx+54. To find a and b, set up a system to be solved.

$$a+b=-21 ab=-5\times 54=-270$$

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 -270.

$$1,-270 2,-135 3,-90 5,-54 6,-45 9,-30 10,-27 15,-18$$

Calculate the sum for each pair.

$$1-270=-269 2-135=-133 3-90=-87 5-54=-49 6-45=-39 9-30=-21 10-27=-17 15-18=-3$$

The solution is the pair that gives sum -21.

$$a=9 b=-30$$

Rewrite -5x^{2}-21x+54 as \left(-5x^{2}+9x\right)+\left(-30x+54\right).

$$\left(-5x^{2}+9x\right)+\left(-30x+54\right)$$

Factor out -x in the first and -6 in the second group.

$$-x\left(5x-9\right)-6\left(5x-9\right)$$

Factor out common term 5x-9 by using distributive property.

$$\left(5x-9\right)\left(-x-6\right)$$

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.

$$-5x^{2}-21x+54=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-5\right)\times 54}}{2\left(-5\right)}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\left(-5\right)\times 54}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-21\right)±\sqrt{441+20\times 54}}{2\left(-5\right)}$$

Multiply 20 times 54.

$$x=\frac{-\left(-21\right)±\sqrt{441+1080}}{2\left(-5\right)}$$

Add 441 to 1080.

$$x=\frac{-\left(-21\right)±\sqrt{1521}}{2\left(-5\right)}$$

Take the square root of 1521.

$$x=\frac{-\left(-21\right)±39}{2\left(-5\right)}$$

The opposite of -21 is 21.

$$x=\frac{21±39}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{21±39}{-10}$$

Now solve the equation x=\frac{21±39}{-10} when ± is plus. Add 21 to 39.

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

Divide 60 by -10.

$$x=-6$$

Now solve the equation x=\frac{21±39}{-10} when ± is minus. Subtract 39 from 21.

$$x=-\frac{18}{-10}$$

Reduce the fraction \frac{-18}{-10} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -6 for x_{1} and \frac{9}{5} for x_{2}.

$$-5x^{2}-21x+54=-5\left(x-\left(-6\right)\right)\left(x-\frac{9}{5}\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-5x^{2}-21x+54=-5\left(x+6\right)\left(x-\frac{9}{5}\right)$$

Subtract \frac{9}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-5x^{2}-21x+54=-5\left(x+6\right)\times \frac{-5x+9}{-5}$$

Cancel out 5, the greatest common factor in -5 and 5.

$$-5x^{2}-21x+54=\left(x+6\right)\left(-5x+9\right)$$

Evaluate

$$-\left(5x-9\right)\left(x+6\right)$$