How to factor this trinomial 5x^2+3x-14?

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samhouston eNotes educator | Certified Educator

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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:

1, -14

14, -1

2, -7

7, -2

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)

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Rylan Hills eNotes educator | Certified Educator

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We have to factorize 5x^2+3x-14

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)

 

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giorgiana1976 | Student

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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