Question

Find γ so that one of the zeros of the quadratic polynomial x^2 - γx +(γ - 1) is double the other

Answer 1

The y value would be -1.5 in order to get roots of -.5 and -1, one of which is double the other.

In order to find this, we can set the roots values at root#1 = a and root#2 = 2a. This is because we know the second should be twice the first one. Now, knowing the way that factoring works, we know that the two roots added together must equal the middle term, and multiplied together must equal the final term.

a + 2a = -y

a(2a) = y - 1

Now we can solve the first one for y and plug that into the second equation.

a + 2a = -y

3a = -y

-3a = y

Place into the next equation.

a(2a) = y - 1

a(2a) = -3a - 1

2a^2 = -3a - 1

2a^2 + 3a + 1 = 0

Now we can use this quadratic to find the roots. These roots would be -1 and -.5. We can now use this in parenthesis to find the standard form of the equation.

(x + 1)(x + .5)

x^2 + 1.5x + .5

Now that we have this, we can solve for the y using either the middle or end term. We'll use the middle term for the purpose here.

1.5x = -yx ----> divide by x

1.5 = -y ---> divide by -1

y = -1.5