To solve this equation with the Zero Product Property, we must first factor the left-hand side. To do that, we can use a generic rectangle and a diamond problem. We know that x^2 and 33 on the left-hand side goes into the lower left and upper right corner of the generic rectangle.
To fill in the remaining two corners, we need to find two x-terms that sum to 14x and have a product of 33x^2.
Notice that the product and sum are both positive. This means both factors must be positive.
|c|c|c|c|c|
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Product & ax(bx) & ax+bx & Sum & 14x? [0.2em]
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33x^2 & 3x(11x) & 3x+11x & 14x & âś“ [0.3em]
When one factor is 3x and the other is 11x we have a product of 33x^2 and a sum of 14x. 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+14x+33=0
⇕
(x+3)(x+11)=0
Now we can solve the equation with the Zero Product Property.
Let's now try a different method. When completing the square we look at the x-term, 14x. The coefficient is 14, which means that we need to add ( 142)^2 to both sides of the equation to complete the square.
