Date: Mon, 29 Mar 1999 18:36:28 -0600
Sender: Ken

#1 a2 - 10a = 2a - 36

#2 t(t - 5) = 5 (t - 5)

#3. (x + 4)(x - 3) = 8

Thanks.

Hi Ken,

I presume that you wish to solve these equations.  (I.e. determine what value(s) of x will make the equations true.)

1. a2 - 10a = 2a - 36
This is a quadratic equation since the highest power of x is the 2nd power. The usual strategy for solving a quadratic equation is to 1st get a zero on one side. a2 - 12a + 36 = 0. Then you can factorthis equation as (a - 6)(a - 6) = 0. Since the product of two binomials is 0 then at least one of them has to be 0. So a - 6 = 0 and a = 6.

2. t(t - 5) = 5(t - 5).
Again, moving both terms to the left side of the equation, t(t - 5) - 5(t - 5) = 0. One of the first steps in factoring is to look for a common factor. Here (t - 5) is a common factor so the equation becomes (t - 5)(t - 5) = 0. Thus, as in 1., t = 5.

3. (x + 4)(x - 3) = 8.
Notice that this is also a quadratic equation and there are, at most, 2 values of x which satisfy the equation. You can proceed as in 1.(move the 8 to the left side, expand and then factor the quadratic) but this is a problem where "guess-and-check" might work. If we can find 2 values of x by inspection we are finished. We see that the product of the 2 binomials on the Left Side is 8. The integer factors of 8 are 1,8,2,4,-1,-8,-2 and-4 so we can try these values for (x + 4).
 x + 4 = 1: Gives x = 5 and thus x - 3 = 2 so (x + 4)(x - 3) = 2 x + 4 = 8: Gives x = 4 and thus x - 3 = 1 so (x + 4)(x - 3) = 8 x + 4 = 2: Gives x = 6 and thus x - 3 = 3 so (x + 4)(x - 3) = 6 x + 4 = 4: Gives x = 0 and thus x - 3 = -3 so (x + 4)(x - 3) = -12 x + 4 = -1: Gives x = -5 and thus x - 3 = -5 so (x + 4)(x - 3) = 8
We now have two values x = 4, -5 that give (x + 4)(x - 3) = 8 and thus we are finished.
Cheers
Jack and Penny
Go to Math Central