Find a second degree polynomial with roots 1/2 and -4 and passes through the point (-1, 6).?

Question

Write your answer in the form f(x) = ax^2 + bx + c.

Answer 1

Roots 1/2 and -4 means the equation is a polynomial of the form: (x - 1/2)(x + 4)

The full form of the equation is:

f(x) = a (x - 1/2)(x + 4)

... where "a" is some constant.

Now all you need to do is substitute for f(-1)=6, and solve for a:

f(x) = a (x - 1/2)(x + 4)
6 = a (-1 -1/2)(-1 + 4)
6 = a (-3/2)(3)
2 = (-3/2) a
-4/3 = a

That gives you the equation:

f(x) = (-4/3) (x - 1/2)(x + 4)

Then you multiply it out to get the form that is requested for the answer.

f(x) = (-4/3) (x - 1/2)(x + 4)
f(x) = (-4/3) (x^2 - (1/2)x + 4x - 2)
f(x) = (-4/3) (x^2 + (7/2)x - 2)
f(x) = (-4/3)x^2 - (14/3)x + 8/3

You can check it by seeing if f(x)=0 when x=1/2 or -4, and also if f(-1)=6

f(x) = (-4/3)x^2 - (14/3)x + 8/3
f(1/2) = (-4/3)(1/2)^2 - (14/3)(1/2) + 8/3
f(1/2) = -1/3 - 7/3 + 8/3
f(1/2) = 0

f(x) = (-4/3)x^2 - (14/3)x + 8/3
f(-4) = (-4/3)(-4)^2 - (14/3)(-4) + 8/3
f(-4) = -64/3 + 56/3 + 8/3
f(-4) = 0

f(x) = (-4/3)x^2 - (14/3)x + 8/3
f(-1) = (-4/3)(-1)^2 - (14/3)(-1) + 8/3
f(-1) = -4/3 + 14/3 + 8/3
f(-1) = 18/3
f(-1) = 6

Answer 2

Let's start assuming you know what your polynomial already is, and your task is to factor this polynomial into two terms. What do you do? Well, you equate it to zero, and find the roots of the polynomial. Once you find these roots (say they're m and n) then you know the factored form of the polynomial is (x-m)(x-n).

So what we're starting with is (x-m)(x-n) right? Because you're given the two roots. Actually, becuase of the other piece of data ( the point you're given) our quadratic needs another parameter to fix: it will be the coefficient of the x^2 term: a.


So your polynomial is a(x-1/2)(x+4) = ax^2 - (7/2)ax + 2a

so f(x) = ax^2 - (7/2)ax + 2a

But what's a?
Well, just sub in (-1,6) into the equation and solve for a

6 = a(-1)^2 -(7/2)(-1)a -2a

Just solve for a, and you'll have your whole answer.

Answer 3

To add to what McFate stated, plug -1 for all x letters in the function f(x) = a (x - 1/2)(x + 4) and simplify.

This will lead you to the answer.

Guido