Question

If the roots of the quadratic equation \(x^{2}-2 \mathrm{kx}+2 \mathrm{k}^{2}-4=0\) are real, then the range of the values of \(\mathrm{k}\) is (1) \([-2,2]\) (2) \([-\infty,-2] \cup[2, \infty]\) (3) \([0,2]\) (4) None of the above

Short answer

Answer: (1) \([-2,2]\)

Step-by-step solution

Identify the coefficients of the quadratic equation

The given quadratic equation is \(x^2 - 2kx + 2k^2 - 4 = 0\). Comparing it with the standard form, \(ax^2 + bx + c = 0\), we have: a = 1 b = -2k c = 2k^2 - 4

Calculate the discriminant

Now we'll calculate the discriminant using the formula Δ = b² - 4ac: Δ = (-2k)² - 4(1)(2k^2 - 4) Δ = 4k^2 - 8k^2 + 16

Test the discriminant condition for real roots

For the roots to be real, the discriminant should be greater than or equal to zero. So we have: Δ >= 0 4k^2 - 8k^2 + 16 >= 0

Solve the inequality for k

Now, we'll solve the inequality for k. -4k^2 + 16 >= 0 4k^2 <= 16 k^2 <= 4 k >= -2 and k <= 2 The above solution tells us that the range of values of k for which the roots of the equation are real is [-2, 2]. So, the correct answer is: (1) \([-2,2]\).

Key concepts

Discriminant of Quadratic Equation

Understanding the discriminant of a quadratic equation is crucial when determining the nature of the roots. Discriminant, usually represented by 'Δ', is calculated from the coefficients of the quadratic equation of the form ax^2 + bx + c = 0. The formula for the discriminant is Δ = b² - 4ac.

The value of Δ provides vital information about the roots:

• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has one real root, also called a repeated or double root.
• If Δ < 0, the equation has two complex conjugate roots, and no real roots.

For the equation in the exercise, we look for real roots, hence we require Δ ≥ 0. Simplifying the resulting inequality helps us find the condition for 'k' that makes the discriminant non-negative, ensuring real roots.

Solving Quadratic Inequalities

Solving quadratic inequalities is similar to finding the roots of a quadratic equation but involves inequality symbols instead of an equals sign. These inequalities are generally in the form ax^2 + bx + c > 0 or ax^2 + bx + c < 0. The process includes finding the discriminant and then determining the intervals of the variable where the inequality holds true.

This often leads to a range of solutions rather than specific values. In our problem, after finding the discriminant, we convert the equation into the inequality -4k^2 + 16 ≥ 0 and proceed to solve for 'k'. Factoring or completing the square can be helpful techniques for solving such inequalities. By doing so, we identify the range of k that produces real roots for the quadratic equation.

Quadratic Equation Coefficients

The coefficients of a quadratic equation are the numerical factors of the terms with variables. For an equation in standard form, ax^2 + bx + c = 0, 'a', 'b', and 'c' are the coefficients, where 'a' is the coefficient of x^2 (quadratic term), 'b' is the coefficient of x (linear term), and 'c' is the constant term. These coefficients dictate not only the shape and position of the parabola on a graph but also the solutions to the equation.

In the provided exercise, a = 1, b = -2k, and c = 2k^2 - 4. Changes in 'k' impact the value of 'b' and 'c', which in turn affect the discriminant and the possibility of real roots. Understanding the relationship between the coefficients and the discriminant is essential to master the concepts behind quadratic equations and their solutions.