SOLUTION: I know its got to be simple, but I just don't understand quadratic equations. 12x^+5x-3=0 ^=squared

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Question 119027This question is from textbook U of Phoenix elementry and intermediate algebra
: I know its got to be simple, but I just don't understand quadratic equations. 12x^+5x-3=0 ^=squared This question is from textbook U of Phoenix elementry and intermediate algebra

Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
12x^2+5x-3=0
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Solved by graphing:
graph%28400%2C300%2C-10%2C10%2C-10%2C10%2C12x%5E2%2B5x-3%29
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Solved by factoring:
12x^2+9x-4x-3=0
3x(4x+3)-(4x+3)=0
(4x+3)(3x-1)=0
4x+3=0 or 3x-1=0
x = -3/4 or x = 1/3
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Solved by the quadratic formula:
x = [-5 +- sqrt(5^2-4*12*-3)]/(2*12)
x = [-5 +- sqrt(169)]/24
x = [-5 +- 13]/24
x = [-5+13]/24 or x = [-5-13]/24
x = 1/3 or x = -3/4
===========================
Cheers,
Stan H.


Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
12x%5E2%2B5x-3=0

This particular quadratic is factorable, meaning that the roots of the equation are rational numbers, although this one takes a bit of work to find the factors.

A quicker method would be to use the quadratic formula and work backwards to find the factors. Here's the quadratic formula:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+, where a is the coefficient on the x%5E2 term, b is the coefficient on the x term, and c is the constant.

In this problem:

. Now we are certain that the roots of the equation will be rational because 169 is a perfect square, namely 13%5E2.

Now our two answers will be x%5B1%5D=%28-5%2B13%29%2F24=8%2F24=1%2F3 and x%5B2%5D=%28-5-13%29%2F24=-18%2F24=-%283%2F4%29.

Since x=1%2F3, we can say 3x-1=0, and since x=-%283%2F4%29 we can say 4x%2B3=0. These are the two factors of 12x%5E2%2B5x-3, as you will see if you use FOIL to multiply %283x-1%29%284x%2B3%29. If either one of those factors is equal to zero, the entire expression is equal to zero.

Perhaps it bothers you that there are two answers. Take a look at a graph of the function f%28x%29=12x%5E2%2B5x-3.

graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C12x%5E2%2B5x-3%29

See where the graph crosses the x-axis at two points? Each of these points is a zero of the function f%28x%29=12x%5E2%2B5x-3 or a root of the equation 12x%5E2%2B5x-3=0.

Hope this helps,
John