Finding the Value of k to Minimize the Sum of Squares of Roots

The problem asks us to find the value of a constant 'k' such that the sum of the squares of the roots of the given quadratic equation is as small as possible. The given quadratic equation is \(2x^2 - 2(k - 2)x - (k + 1) = 0\).

Understanding Quadratic Equation Roots

A general quadratic equation is given by \(ax^2 + bx + c = 0\). If \(\alpha\) and \(\beta\) are the roots of this equation, we have the following relationships:

• Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
• Product of roots: \(\alpha \beta = \frac{c}{a}\)

The sum of the squares of the roots is given by the identity: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta\).

Applying Root Properties to the Given Equation

For the equation \(2x^2 - 2(k - 2)x - (k + 1) = 0\), we identify the coefficients:

• \(a = 2\)
• \(b = -2(k - 2)\)
• \(c = -(k + 1)\)

Now, we can find the sum and product of the roots in terms of k:

• Sum of roots: \(\alpha + \beta = -\frac{-2(k - 2)}{2} = \frac{2(k - 2)}{2} = k - 2\)
• Product of roots: \(\alpha \beta = \frac{-(k + 1)}{2}\)

Calculating the Sum of Squares of Roots

Next, we substitute these expressions for the sum and product of roots into the formula for the sum of squares:

\(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta\)

\(\alpha^2 + \beta^2 = (k - 2)^2 - 2\left(\frac{-(k + 1)}{2}\right)\)

Let's simplify this expression:

\(\alpha^2 + \beta^2 = (k^2 - 4k + 4) + (k + 1)\)

\(\alpha^2 + \beta^2 = k^2 - 4k + 4 + k + 1\)

\(\alpha^2 + \beta^2 = k^2 - 3k + 5\)

So, the sum of the squares of the roots is a quadratic function of k: \(S(k) = k^2 - 3k + 5\).

Minimizing the Sum of Squares

We want to find the value of k that minimizes the function \(S(k) = k^2 - 3k + 5\). This is a quadratic function in the form \(Ak^2 + Bk + C\) with \(A = 1\), \(B = -3\), and \(C = 5\). Since the coefficient of \(k^2\) (A = 1) is positive, the parabola opens upwards, meaning it has a minimum value at its vertex.

The k-coordinate of the vertex of a parabola \(Ak^2 + Bk + C\) is given by the formula \(k = -\frac{B}{2A}\).

Using this formula for \(S(k) = k^2 - 3k + 5\):

\(k = -\frac{(-3)}{2 \times 1}\)

\(k = \frac{3}{2}\)

This value of k corresponds to the minimum value of the sum of the squares of the roots.

Conclusion

The value of k for which the sum of the squares of the roots of the equation \(2x^2 - 2(k - 2)x - (k + 1) = 0\) is minimum is \(k = \frac{3}{2}\).

Revision Table: Quadratic Equation Roots

Additional Information: Minimizing Quadratic Functions

When you have a quadratic function in the form \(f(x) = Ax^2 + Bx + C\), its graph is a parabola. If \(A > 0\), the parabola opens upwards, and the minimum value occurs at the vertex. If \(A < 0\), the parabola opens downwards, and the maximum value occurs at the vertex.

The vertex can be found using the formula for the x-coordinate \(x = -\frac{B}{2A}\). This is a fundamental concept in algebra and calculus for finding the extremum (minimum or maximum) of a quadratic function. In our problem, the sum of squares of roots \(S(k)\) is a quadratic function of \(k\), and since the coefficient of \(k^2\) is positive, we were looking for its minimum value.