## Quadratic Database

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[1000] Solve: 2x^{2}+10x-6 = 5x^{2}+4x+7

**Solution:**First we subtract
5x^{2}+4x+7 from both sides of the
equation.

2x^{2}+10x-6-(5x^{2}+4x+7) =
5x^{2}+4x+7-(5x^{2}+4x+7)

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

2x^{2}+10x-6-5x^{2}-4x-7 = 0

We collect like terms.

-3x^{2}+6x-13 = 0

We can now use the
quadratic formula with
a = -3, b = 6, c = -13.

But, the discriminant
= 6^{2}-4(-3)(-13) = -120 is less than zero. So this problem has
no solution.

[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)

6x^{2}+11x-10 =
3x^{2}+10x-8

Then we subtract 3x^{2}+10x-8
from both sides and collect like terms.

6x^{2}+11x-10 =
3x^{2}+10x-8

6x^{2}+11x-10-(3x^{2}+10x-8) =
3x^{2}+10x-8-(3x^{2}+10x-8)

6x^{2}+11x-10-3x^{2}-10x+8 = 0

3x^{2}+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.

[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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