a Stanton is talking about the Zero Product Property. When x equals 5 or -2, the left-hand side will equal 0. However, a solution to an equation has to make the left-hand side and right-hand side equal. This is not the case for the given equation and proposed solutions.
( 5-5)( 5+2)? =-6 & ⇔ 0 ≠-6 *
( -2-5)( -2+2)? =-6 & ⇔ 0 ≠-6 *
As we can see, the proposed solutions does not result in true statements. Therefore, these values of x cannot be solutions to the equation.
b To rewrite the equation, we must make sure that the right-hand side equals 0.
Now the right-hand side is 0. To factorize the left-hand side, we can use a generic rectangle and a diamond problem. We know that x^2 and -4 goes into the lower left and upper right corners of the generic rectangle.
To fill in the remaining two corners, we need two x-terms that have a sum of -3x and a product of - 4x^2.
Notice that the product is negative. This means one factor must be positive and the other must be negative.
|c|c|c|r|c|
[-1em]
Product & ax(bx) & ax+bx & Sum & -3x? [0.2em]
[-1em]
-4x^2 & - x(4x) &- x+4x& 3x & * [0.1em]
-4x^2 & - 4x(x) & -4x+x& -3x & ✓ [0.1em]
When one term is - 4x and the other is x, we have a product of -4x^2 and a sum of -3x. Now we can complete the diamond and generic rectangle.
To factor the left-hand side we add each side of the generic rectangle and multiply the sums.
x^2-3x-4=0
⇓
(x+1)(x-4)=0
c Since the right-hand side now equals zero, we can solve the equation with the Zero Product Property.
