-7+5=-2 and I got -2/2.
That was supposed to be (-7+5)/4 to get -0.5, right?
Correct.
Btw, you may report x = -1/2, even though your calculator displays -0.5 (
CASE-IN-POINT: Some students who consistently avoid working with small fractions eventually lose their ability to work with small fractions.)
:idea: You can always check your candidates for solutions by substituting them (one at a time) into the original equation, followed by doing all of the resulting arithmetic, to ensure that they work by demonstrating that you end up with a true statement (like 0 = 0 or 22/7 = 22/7 or -5=-5).
The Zero-Product Property tells us that anytime you get zero after multiplying things together, at least one of those things must be zero. In other words, it is impossible to get zero as a product when multiplying non-zero numbers together.
EGs:
Given a*b = 0
We can tell just by looking at this equation that either a=0 or b=0 or both a and b equal zero.
We can't be sure which of these three scenarios is true without further information, but the Zero-Product Property tells us that ONE OF THOSE POSSIBILITIES must be true. So, we answer a = 0 OR b = 0 OR both equal zero to cover all possibilities.
Given (x - 4)(3x + 7) = 0
We can tell just by looking that IF the factor x - 4 = 0 then the product on the left side will equal zero because (0)(3x + 7) = 0.
Same result IF factor 3x + 7 = 0 because (x - 4)(0) = 0.
The examples above show why zero is such a powerful number.
Multiplication by zero always results in zero.
What are the solutions to the equation (x - 4)(3x + 7) = 0 ?
Well, on the left side of the equation, the two unknown numbers (expressed above as x-4 and 3x+7) must multiply together to make zero. We already know that if just
one of the two numbers is zero, then it does not matter what the other number is. Right?
Let's
assume, one by one, that either factor
is zero. Either of them could be; after all, their values are unknown to us at this point.
So, we apply the Zero-Product Property and write:
x - 4 = 0
OR
3x + 7 = 0
The solutions to these two equations (obtained by assuming that each factor is zero) will be the same as the solutions to the original equation.
This is because a value for the variable x=4 makes the expression x-4 zero AND a value x=-7/3 causes the number 3x+7 to evaluate to zero.
Check it out.
(x - 4)(3x + 7) = 0
(4 - 4)(12 + 7) = 0
(0)(19) = 0
0 = 0
(x - 4)(x*3 + 7) = 0
(-7/3 - 4)(-7/3 * 3 + 7) = 0
(-7/3 - 12/3)(-7 + 7) = 0
(-19/3)(0) = 0
0 = 0
Both assumed solutions check, and we know that a quadratic equation may not have more than two different solutions. Hence, the solutions to the original equation are:
x = 4 OR x = -7/3
Now, try using the Zero-Product Property on your exercise, and check that you get the same results as you did using the Quadratic Formula. Which method is easier for you? :cool: (
DISCLAIMER: When an instructor specifically instructs to use the Quadratic Formula for a particular exercise, then one must demonstrate the Quadratic Formula.)
