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# sum and product of roots

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If (2x + 5)(x + 3) = 14, find the sum of the possible values of x.

Mar 14, 2021

### 1+0 Answers

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This is a quadratic, and to find x in a quadratic, you can use the quadratic formula.

First, expand the equation:

(2x+5)(x+3)=14

2x^2+6x+5x+15=14

2x^2+11x+15=14

2x^2+11x+1=0

Now, the equation is in

ax^2+bx+c=0

form.

This tells us that

a=2

b=11

c=1

The quadratic formula is:

x=(-b +- sqrt(b^2-4ac))/2a

Now, plugging in the variables, you get:

x=(-11 +- sqrt(11^2-4*2*1))/2*2

x=(-11 +- sqrt(121-8))/4

x=(-11 +- sqrt(113))/4

Now you can find the two possible values of x. When you let the equation have a plus sign in front of the sqrt, you get:

x=(-11+sqrt(113))/4

When you let the equation have a minus sign in front of the sqrt, you get:

x=(-11-sqrt(113))/4

Now you have your two possible values of x.

The question asks for the sum of the possible values of x, so all you have to do is add the two values of x.

(-11+sqrt(113)-11-sqrt(113))/4

(-11-11)/4

-22/4

ANSWER FOR THE SUM OF THE ROOTS: -22/4 OR -5.5

I saw in the title that you wanted the product of the roots also, even though you didn't include that in the actual problem, so here it is.

All you have to do is multiply the roots.

((-11+sqrt(113))/4)*((-11-sqrt(113))/4)

Doing some math that you probably know how to do, you get:

1/2

ANSWER FOR THE PRODUCT OF THE ROOTS: 1/2 OR 0.5

All of the work above is the LONG way of doing it. There is actually an easier way that you can use after you understand how it works.

To find the sum of the roots, all you have to do is:

(-b/a)

To find the product of the roots, all you have to do is:

(c/a)

Using this gives us the same answer:

(-b/a)

-11/2

-5.5

(c/a)

1/2

0.5

Hopefully this helps, this is my third answer :)

Note: I do all my calculations by hand so my method should be correct but you should double-check my work.

Mar 14, 2021