Question

How do you evaluate `(8x + 4) ( 10x - 3) = 0`?

Answer 1

Either set each binomial equal to 0, or use the FOIL method and the quadratic formula, to determine `x=3/10` and `x=-1/2`

Explanation:

There are two methods of doing this, a simple one and a difficult one. We will open with the simple one. The equation is only satisfied if the left hand side is equal to 0, which will happen if either binomial is equal to zero. Therefore we can instead solve:

`8x+4 = 0` and `10x-3=0`

Moving the constants to the other side and dividing by the coefficient of x we get:

`x=-1/2 or x=3/10`

This is the simplest means of getting the answer. If you wish to go the longer route of multiplying the binomials and using the quadratic formula, see below:

The method commonly used to multiply these terms is the FOIL method, aka "First, Outer, Inner, Last." What this means is that to do the multiplication we first multiply the first two terms of each binomial (8x, 10x), then the outermost terms (8x, -3), the innermost terms (4, 10x), and finally the last terms (4, -3), and add them all together.

We could also simply take one binomial and multiply both terms therein by each of the terms of the other binomial in turn, but the FOIL method is the better known technique.

`(8x+4)(10x-3) = (8x*10x) + (8x*-3)+(4*10x)+(4*-3) = 80x^2 - 24x + 40x - 12 = 80x^2 + 16x - 12 = 0 = 4(20x^2+4x-3)`

Then, because 0/4 is 0

`-> 20x^2 + 4x -3 = 0`

Now that we have that, we can use the quadratic formula, which states that for a quadratic equation `Ax^2 + Bx+C =0`, the zeroes can be found as follows:

`(-b+- sqrt(b^2-4ac))/(2a)`

In this case...

`-> (-4+-sqrt(4^2-4(20)(-3)))/40 = (-4 +- sqrt (256))/40 = (-4+-16)/40`

`(-4+16)/40 = 12/40 = 6/20 = 3/10, (-4-16)/40 = -20/40 = -1/2`

Thus our zeroes are `x=3/10` and `x=-1/2`