Solve -5x^2-21x+20

$$-5 { x }^{ 2 } -21x+20$$

Factor

$$-\left(5x-4\right)\left(x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -5x^{2}+ax+bx+20. To find a and b, set up a system to be solved.

$$a+b=-21 ab=-5\times 20=-100$$

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 -100.

$$1,-100 2,-50 4,-25 5,-20 10,-10$$

Calculate the sum for each pair.

$$1-100=-99 2-50=-48 4-25=-21 5-20=-15 10-10=0$$

The solution is the pair that gives sum -21.

$$a=4 b=-25$$

Rewrite -5x^{2}-21x+20 as \left(-5x^{2}+4x\right)+\left(-25x+20\right).

$$\left(-5x^{2}+4x\right)+\left(-25x+20\right)$$

Factor out -x in the first and -5 in the second group.

$$-x\left(5x-4\right)-5\left(5x-4\right)$$

Factor out common term 5x-4 by using distributive property.

$$\left(5x-4\right)\left(-x-5\right)$$

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.

$$-5x^{2}-21x+20=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-5\right)\times 20}}{2\left(-5\right)}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\left(-5\right)\times 20}}{2\left(-5\right)}$$

Multiply -4 times -5.

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

Multiply 20 times 20.

$$x=\frac{-\left(-21\right)±\sqrt{441+400}}{2\left(-5\right)}$$

Add 441 to 400.

$$x=\frac{-\left(-21\right)±\sqrt{841}}{2\left(-5\right)}$$

Take the square root of 841.

$$x=\frac{-\left(-21\right)±29}{2\left(-5\right)}$$

The opposite of -21 is 21.

$$x=\frac{21±29}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{21±29}{-10}$$

Now solve the equation x=\frac{21±29}{-10} when ± is plus. Add 21 to 29.

$$x=\frac{50}{-10}$$

Divide 50 by -10.

$$x=-5$$

Now solve the equation x=\frac{21±29}{-10} when ± is minus. Subtract 29 from 21.

$$x=-\frac{8}{-10}$$

Reduce the fraction \frac{-8}{-10} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and \frac{4}{5} for x_{2}.

$$-5x^{2}-21x+20=-5\left(x-\left(-5\right)\right)\left(x-\frac{4}{5}\right)$$

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

$$-5x^{2}-21x+20=-5\left(x+5\right)\left(x-\frac{4}{5}\right)$$

Subtract \frac{4}{5} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-5x^{2}-21x+20=-5\left(x+5\right)\times \frac{-5x+4}{-5}$$

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

$$-5x^{2}-21x+20=\left(x+5\right)\left(-5x+4\right)$$

Evaluate

$$-\left(5x-4\right)\left(x+5\right)$$