The quadratic equation 2qx(x+1)-8x^2-1, where q is a constant, has no real roots. Find the range of values of q.
Answer given is -2<q<4
I will assume that you meant the equation to be
2qx(x+1) - 8x^2 - 1 = 0
2qx^2 + 2qx - 8x^2- 1 = 0
x^2(2q - 8) + 2qx - 1 = 0
a quadratic with
a = 2q-8
b=2q
c=-1
to have no real roots, b^2 - 4ac < 0
4q^2 - 4(2q-8)(-1) < 0
4q^2 + 8q - 32 < 0
q^2 + 2q - 8 < 0
(q+4)(q-2) < 0
critical values: q = -4, q = 2
so -4 < q < 2
notice my signs are opposite of the given answer.
I can't find any errors in my solution, perhaps you can if there are any.
Let me know
The equation is 2qx(x+1)=-8x^2 -1
that's not what you had, so make the necessary changes in my solution.
I'm really sorry, but how?
To find the range of values of q for which the quadratic equation has no real roots, we can use the discriminant.
The discriminant is the part of the quadratic formula under the square root sign, which helps determine the nature of the roots. For a quadratic equation in the form ax^2 + bx + c = 0, the discriminant is given by b^2 - 4ac.
In this case, the quadratic equation is 2qx(x+1) - 8x^2 - 1. We need to rearrange it in the standard form ax^2 + bx + c = 0, so let's expand and simplify the equation:
2qx^2 + 2qx - 8x^2 - 1 = 0
Combining like terms, we have:
(2q - 8)x^2 + (2q)x - 1 = 0
Comparing this with the standard form of the quadratic equation, we can see that a = (2q - 8), b = 2q, and c = -1.
Now, let's calculate the discriminant:
Discriminant = b^2 - 4ac = (2q)^2 - 4(2q - 8)(-1)
Simplifying this expression:
Discriminant = 4q^2 + 32q + 32
For the quadratic equation to have no real roots, the discriminant should be negative (less than zero).
Therefore, we have the inequality:
4q^2 + 32q + 32 < 0
To solve this inequality for q, we can factorize the quadratic expression:
(q + 2)(q - 4) < 0
To determine the range of q values, we need to consider the signs of the expressions (q + 2) and (q - 4) for different intervals.
Case 1: (q + 2) > 0 and (q - 4) < 0
This means q > -2 and q < 4.
Case 2: (q + 2) < 0 and (q - 4) > 0
This means q < -2 and q > 4 (which is not possible).
Therefore, the possible range of values for q is:
-2 < q < 4
Hence, the answer to the question is -2 < q < 4.