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[1000] Solve: 2x²+10x-6 = 5x²+4x+7

Solution:First we subtract 5x²+4x+7 from both sides of the equation.

2x²+10x-6-(5x²+4x+7) = 5x²+4x+7-(5x²+4x+7)

We use the distributive law to expand the left hand side. Of course, the right hand side is zero.

2x²+10x-6-5x²-4x-7 = 0

We collect like terms.

-3x²+6x-13 = 0

We can now use the quadratic formula with a = -3, b = 6, c = -13.
But, the discriminant = 6²-4(-3)(-13) = -120 is less than zero. So this problem has no solution.

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[1001] Solve: (3x-2)(2x+5)=(3x-2)(x+4)

First, we use FOIL to get rid of the brackets.

(3x-2)(2x+5) = (3x-2)(x+4)

6x²+11x-10 = 3x²+10x-8

Then we subtract 3x²+10x-8 from both sides and collect like terms.

6x²+11x-10 = 3x²+10x-8

6x²+11x-10-(3x²+10x-8) = 3x²+10x-8-(3x²+10x-8)

6x²+11x-10-3x²-10x+8 = 0

3x²+x-2 = 0

We can now use the quadratic formula with a = 3, b = 1, c = -2.
We find the lower root is -1 and the upper root is 2/3.

Alternative solution: The really smart (a.k.a. observant) student would have noticed (3x-2) on both sides of the equation. If 3x - 2 = 0, then automatically the left side equals the right side since we would have 0 = 0.
The solution to 3x - 2 = 0 is, of course, x = 2/3, the upper root we got using the quadratic formula.
If 3x - 2 is not zero, then we can divide both sides by (3x - 2) to get 2x + 5 = x + 4. Using the strategy to solve linear problems, we get x = -1, the lower root we got using the quadratic formula.

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[1002] Solve:

Solution: Since we're dealing with fractions here, the first thing we need to do is find out what the answer can't be, because we don't want to be dividing by zero. That means that neither 8x + 16 nor -4x + 16 can be equal to zero. So we can scratch out x = -2 and x = 4 as solutions.

Now that we have that taken care of, we can get on with the business of finding out what the solutions are. The first thing we do is cross-multiply to get rid of the fractions. We get:

(x+2)(-4x+16) = (x-1)(8x+16)

There are 2 ways we can tackle this problem now: the smart way and the not-so-smart way. The not-so-smart way is to use FOIL to get rid of the brackets on both sides, bring everything over to one side and use the quadratic formula to find the roots. If we did that (feel free to try it if you want), we would find out that one root is -2 and the other is 2. But since x = -2 is not a solution (remember?) the only solution is x = 2.

The intelligent solution is to see that (8x+16) = 8(x+2) on the right hand side of the equation. We now have:

(x+2)(-4x+16) = (x-1)(8x+16)

(x+2)(-4x+16) = 8(x-1)(x+2)

And since (x+2) can't be zero (because x can't be -2), we can divide both sides by (x+2):

(x+2)(-4x+16) = 8(x-1)(x+2)

-4x+16 = 8(x-1)

Now we are left with solving a basic problem involving brackets.

-4x+16 = 8(x-1)

-4x+16 = 8x-8

-4x+16-8x = 8x-8-8x

-12x+16 = -8

-12x+16-16 = -8-16

-12x = -24

-12x/-12 = -24/-12

x = 2

We have the same solution as we would have gotten using the quadratic formula, but with a bit less work.

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