Solve the equation negative 4x squared plus 20x minus 24 equals 0 by factoring.

Step-by-step solution

1. 1

   Deal with the negative leading coefficient before anything else. Trying to factor $-4x^2+20x-24$ directly means juggling a negative $a$ and a common factor at the same time. All three coefficients are divisible by $-4$, so pull it out:

   $-4x^2+20x-24=-4\left(x^2-5x+6\right)$

   Dividing each term by $-4$ flips every sign: $-4x^2\div(-4)=x^2$, $20x\div(-4)=-5x$, $-24\div(-4)=6$.

2. 2

   Factor the monic quadratic inside. Two numbers with product $6$ and sum $-5$ are $-2$ and $-3$:

   $x^2-5x+6=(x-2)(x-3)$

   So the equation is $-4(x-2)(x-3)=0$.

3. 3

   Divide by the non-zero constant. Unlike dividing by a variable, dividing by $-4$ is completely safe because $-4$ can never be zero, so no root is lost:

   $(x-2)(x-3)=0$

4. 4

   Apply the zero product property.

   $x-2=0\ \text{or}\ x-3=0\ \Longrightarrow\ x=2\ \text{or}\ x=3$

5. 5

   Verify in the original equation, negative sign and all.

   $-4(2)^2+20(2)-24=-16+40-24=0\quad\checkmark$

   $-4(3)^2+20(3)-24=-36+60-24=0\quad\checkmark$