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5x^2 + 3x - 14
First, set up your two binomials. Since the middle term is positive and the last term is negative, you know one of the binomials contains addition and the other binomial contains subtraction.
(___ + ___)(___ - ___)
The coefficient of the first term (5) is prime, so the first terms in the binomials are 5x and x.
(5x + ___)(x - ___)
The last term (-14) can be factored a few ways:
Now try out each combination. Using FOIL, find the combination that gives you the middle term 3x.
(5x + 1)(x - 14) middle term = 69x no
(5x + 14)(x - 1) middle term = 9x no
(5x + 2)(x - 7) middle term = -33x no
(5x + 7)(x - 2) middle term = -3x
That last combination was close, except we need the middle term to be postive 3x, not negative 3x. So we switch the + and - signs and try again.
(5x - 7)(x + 2) middle term = 3x
Use FOIL to check.
5x^2 + 10x - 7x - 14
5x^2 + 3x - 14
Since this was your original trinomial, it has been factored correctly.
Solution: (5x - 7)(x + 2)
We have to factorize 5x^2+3x-14
=> 5x^2 + 10x - 7x - 14
=> 5x(x + 2) - 7(x + 2)
=> (5x - 7)(x + 2)
The factors of 5x^2+3x-14 = (5x - 7)(x + 2)
The trinomial could be factored if we know it's roots:
(x - x1)(x - x2), where x1 and x2 are it's roots.
To factor this trinomial, we'll apply the quadratic formula:
x1 = [-3+sqrt(3^2 - 4*5*(-14))]/2*5
x1 = (-3 + sqrt289)/10
x1 = (-3+17)/10
x1 = 14/10
x2 = (-3-17)/10
x2 = -20/10
x2 = -2
The factored form of the given trinomial is: (x - 14/10)(x + 2).
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