Understanding Quadratic Equations and Real Roots

The problem asks for the number of integer values of $\small k$ for which the quadratic equation $\small x^2 - 4x + k = 0$ has real roots, and both of these roots lie strictly between 0 and 5 (i.e., in the interval $\small (0, 5)$).

For a quadratic equation of the form $\small ax^2 + bx + c = 0$, several conditions must be met for the roots to be real and lie within a specific interval $\small (p, q)$. In our equation, $\small a = 1$, $\small b = -4$, and $\small c = k$. The interval is $\small (p, q) = (0, 5)$. Since $\small a=1$ is positive, the parabola opens upwards.

Conditions for Real Roots

A quadratic equation has real roots if and only if its discriminant, $\small \Delta = b^2 - 4ac$, is greater than or equal to zero.

For $\small x^2 - 4x + k = 0$, the discriminant is:

$\small \Delta = (-4)^2 - 4(1)(k) = 16 - 4k$

For real roots, we must have:

$\small 16 - 4k \ge 0$

$\small 16 \ge 4k$

$\small 4 \ge k$

So, $\small k \le 4$. This is our first condition on $\small k$.

Conditions for Roots within the Interval (0, 5)

For a quadratic equation $\small f(x) = ax^2 + bx + c = 0$ with $\small a > 0$, for both roots to lie in the interval $\small (p, q)$, the following conditions must be satisfied:

• The discriminant $\small \Delta \ge 0$ (for real roots).
• $\small f(p) > 0$.
• $\small f(q) > 0$.
• The x-coordinate of the vertex of the parabola, $\small -b/(2a)$, must lie within the interval $\small (p, q)$, i.e., $\small p < -b/(2a) < q$.

Let's apply these conditions to $\small f(x) = x^2 - 4x + k$ and the interval $\small (0, 5)$:

Condition 1: Discriminant ($\small \Delta \ge 0$)

We already found this condition:

$\small k \le 4$

Condition 2: $\small f(0) > 0$

Substitute $\small x = 0$ into the equation $\small f(x) = x^2 - 4x + k$:

$\small f(0) = (0)^2 - 4(0) + k = k$

For the roots to be in $\small (0, 5)$, we need $\small f(0) > 0$ (as the parabola opens upwards, if $\small f(0) \le 0$, at least one root would be less than or equal to 0).

$\small k > 0$

Condition 3: $\small f(5) > 0$

Substitute $\small x = 5$ into the equation $\small f(x) = x^2 - 4x + k$:

$\small f(5) = (5)^2 - 4(5) + k = 25 - 20 + k = 5 + k$

For the roots to be in $\small (0, 5)$, we need $\small f(5) > 0$ (similarly, if $\small f(5) \le 0$, at least one root would be greater than or equal to 5).

$\small 5 + k > 0$

$\small k > -5$

Condition 4: Vertex in Interval (0, 5)

The x-coordinate of the vertex is given by $\small -b/(2a)$. For $\small x^2 - 4x + k = 0$, the vertex x-coordinate is:

$\small -(-4)/(2 \times 1) = 4/2 = 2$

The condition is that the vertex must lie in $\small (0, 5)$. So, $\small 0 < 2 < 5$. This statement is true. This condition does not impose any further restrictions on the value of $\small k$.

Combining the Conditions for k

We have the following conditions for $\small k$:

• $\small k \le 4$ (from $\small \Delta \ge 0$)
• $\small k > 0$ (from $\small f(0) > 0$)
• $\small k > -5$ (from $\small f(5) > 0$)

We need to find the values of $\small k$ that satisfy all three inequalities simultaneously. Combining $\small k > 0$ and $\small k > -5$, we get $\small k > 0$. Combining $\small k > 0$ and $\small k \le 4$, we get the range for $\small k$ as:

$\small 0 < k \le 4$

Finding Integral Values of k

The problem asks for the number of integral values of $\small k$. The integers $\small k$ that satisfy $\small 0 < k \le 4$ are the integers greater than 0 and less than or equal to 4.

These integers are 1, 2, 3, and 4.

Let's check each value:

• If $\small k=1$: $\small x^2 - 4x + 1 = 0$. $\small \Delta = 16 - 4(1) = 12 > 0$. Roots are $\small x = (4 \pm \sqrt{12})/2 = 2 \pm \sqrt{3}$. $\small \sqrt{3} \approx 1.732$. Roots are $\small 2 - 1.732 = 0.268$ and $\small 2 + 1.732 = 3.732$. Both $\small 0.268$ and $\small 3.732$ are in $\small (0, 5)$. Valid.
• If $\small k=2$: $\small x^2 - 4x + 2 = 0$. $\small \Delta = 16 - 4(2) = 8 > 0$. Roots are $\small x = (4 \pm \sqrt{8})/2 = 2 \pm \sqrt{2}$. $\small \sqrt{2} \approx 1.414$. Roots are $\small 2 - 1.414 = 0.586$ and $\small 2 + 1.414 = 3.414$. Both $\small 0.586$ and $\small 3.414$ are in $\small (0, 5)$. Valid.
• If $\small k=3$: $\small x^2 - 4x + 3 = 0$. $\small \Delta = 16 - 4(3) = 4 > 0$. Roots are $\small x = (4 \pm \sqrt{4})/2 = (4 \pm 2)/2$. Roots are $\small (4-2)/2 = 1$ and $\small (4+2)/2 = 3$. Both 1 and 3 are in $\small (0, 5)$. Valid.
• If $\small k=4$: $\small x^2 - 4x + 4 = 0$. $\small \Delta = 16 - 4(4) = 0$. The equation is $\small (x-2)^2 = 0$. The root is $\small x = 2$ (a repeated root). 2 is in $\small (0, 5)$. Valid.

The integral values of $\small k$ that satisfy all conditions are 1, 2, 3, and 4.

There are 4 such integral values.

Conclusion

The integral values of $\small k$ for which the equation $\small x^2 - 4x + k = 0$ has real roots and both roots lie in the interval $\small (0, 5)$ are 1, 2, 3, and 4.

The number of such integral values is 4.

Revision Table: Quadratic Roots Analysis

Additional Information: Location of Quadratic Roots

Understanding the location of roots for a quadratic equation $\small ax^2 + bx + c = 0$ is crucial for solving problems like this. The sign of $\small a$ determines the direction of the parabola. If $\small a > 0$, the parabola opens upwards; if $\small a < 0$, it opens downwards.

To determine if both roots lie within an interval $\small (p, q)$, besides ensuring real roots ($\small \Delta \ge 0$), we check the value of the function at the interval boundaries $\small p$ and $\small q$ and the position of the vertex.

• If $\small a > 0$:
  • Roots in $\small (p, q)$ requires $\small f(p) > 0$, $\small f(q) > 0$, and $\small p < -b/(2a) < q$.
  • One root in $\small (p, q)$ requires $\small f(p)$ and $\small f(q)$ to have opposite signs, i.e., $\small f(p)f(q) < 0$.
• If $\small a < 0$:
  • Roots in $\small (p, q)$ requires $\small f(p) < 0$, $\small f(q) < 0$, and $\small p < -b/(2a) < q$.
  • One root in $\small (p, q)$ requires $\small f(p)f(q) < 0$.

These conditions help narrow down the possible values of the coefficients based on the desired location of the roots.