the equation (x-1)(x-3)=k has both roots lying between 1 and 3, then number of integral values of k is
since the vertex of (x-1)(x-3) = 0 is at (2,-1), k ≥ -1
or, since
x^2 - 4x + 3-k has discriminant 16-4(3-k) = 4(1+k),
1+k must not be negative
To find the number of integral values of k for the equation (x-1)(x-3) = k, we need to consider the conditions for the roots to lie between 1 and 3.
Let's expand the equation by multiplying the terms:
x^2 - 4x + 3 = k
Now, let's analyze the conditions for the roots to lie between 1 and 3.
1. For the roots to be less than 3:
To satisfy this condition, the discriminant (b^2 - 4ac) must be positive.
Discriminant = (-4)^2 - 4(1)(3 - k)
= 16 - 12 + 4k
= 4k + 4
For the discriminant to be positive: 4k + 4 > 0
Simplifying the inequality: k > -1
Thus, the condition for the roots to be less than 3 is k > -1.
2. For the roots to be greater than 1:
To satisfy this condition, both roots need to be greater than 1. In other words, the equation should not have any real roots less than or equal to 1.
The equation (x-1)(x-3) = k will have real roots if and only if the discriminant is nonnegative.
Discriminant = (-4)^2 - 4(1)(3 - k)
= 16 - 12 + 4k
= 4k + 4
For the discriminant to be nonnegative: 4k + 4 ≥ 0
Simplifying the inequality: k ≥ -1
Thus, the condition for the roots to be greater than 1 is k ≥ -1.
Combining the conditions, we get -1 < k.
To find the number of integral values of k within this range, we can count the number of integers greater than -1. Since there are infinitely many integers greater than -1, the number of integral values of k is infinite.
Therefore, the number of integral values of k for the given equation is infinite.
To find the number of integral values of k, we need to analyze the given equation (x-1)(x-3) = k and determine the conditions for which both roots lie between 1 and 3.
Let's first expand the equation:
(x-1)(x-3) = k
x^2 - 3x - x + 3 = k
x^2 - 4x + 3 = k
For both roots to lie between 1 and 3, we need the discriminant (b^2 - 4ac) to be positive, where a = 1, b = -4, and c = 3.
D = b^2 - 4ac
D = (-4)^2 - 4(1)(3)
D = 16 - 12
D = 4
Since the discriminant is positive (D > 0), it means that the quadratic equation has two distinct real roots.
Furthermore, for both roots to lie between 1 and 3, we also need the individual roots to be greater than 1 and less than 3. So, let's find the roots of the equation:
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a)
x = (4 ± √4) / 2
x = (4 ± 2) / 2
x = 2 ± 1
The roots of the equation are x = 3 and x = 1, which are both within the range of 1 to 3.
Now, let's analyze the range of k values that satisfy the given conditions.
Since we have two roots x = 3 and x = 1, we can substitute these values back into the equation to solve for k:
For x = 3:
(x-1)(x-3) = k
(3-1)(3-3) = k
(2)(0) = k
k = 0
For x = 1:
(x-1)(x-3) = k
(1-1)(1-3) = k
(0)(-2) = k
k = 0
We can observe that when x = 3 and x = 1, k is equal to 0.
Thus, the only integral value of k that satisfies the given conditions is k = 0.
Therefore, the number of integral values of k is 1.