Problem 65 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(k, x^{2}-k x+k+2\) has equal roots?

Step-by-Step Solution

Answer

The values of \(k\) for which the equation \(x^2 - kx + k + 2\) has equal roots are \(k = 6\) and \(k = -2\).

Step 1: Identify coefficients of the quadratic equation

In the given equation, \(x^{2} - kx + k +2\), we have: \(a = 1\) \(b = -k\) \(c = k + 2\)

Step 2: Calculate the discriminant

Now, we find the discriminant using the formula \(D = b^2 - 4ac\): \(D = (-k)^2 - 4(1)(k + 2)\) 2. Set the discriminant equal to zero

Step 3: Set D equal to zero

To find the values of k for which the equation has equal roots, we set D equal to zero: \((k)^2 - 4(k + 2) = 0\) 3. Solve the equation for k

Step 4: Simplify the equation

Expand and simplify the equation: \(k^2 - 4k - 8 = 0\)

Step 5: Factor or use quadratic formula

This equation is not easily factored, so we'll use the quadratic formula: \(k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In our case, \(a = 1\), \(b = -4\), and \(c = -8\). \(k = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(-8)}}{2(1)}\)

Step 6: Simplify further

Computing the square root term: \(k = \frac{4 \pm \sqrt{64}}{2}\) Thus, \(k = \frac{4 \pm 8}{2}\) 4. Find the values of k

Step 7: Compute two possible values of k

Now, we find the two possible values of k: \(k = \frac{4 + 8}{2} = \frac{12}{2} = 6\) And \(k = \frac{4 - 8}{2} = \frac{-4}{2} = -2\) So, the values of k for which the equation \(x^2 - kx + k + 2\) has equal roots are \(k = 6\) and \(k = -2\).

Key Concepts

Discriminant

The discriminant is a vital part of the quadratic equation. It helps us determine the nature of roots that a quadratic equation can have. If you have a quadratic equation in the format \(ax^2 + bx + c = 0\), then the discriminant \(D\) is given by the formula:\[D = b^2 - 4ac\]The discriminant tells us different scenarios about the roots:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), it has two equal real roots (sometimes called repeated or double roots).
• If \(D < 0\), the equation has no real roots, but instead two complex roots.

In the exercise, we used the discriminant to find the condition when the quadratic equation \(x^2 - kx + k + 2\) has equal roots. Therefore, setting \(D = 0\) gives us a necessary relationship between the coefficients for equal roots.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \(x\). Its standard form is given by:\[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants. The coefficient \(a\) must not be zero as it ensures the equation is of degree two. Quadratic equations are crucial because they appear in various practical applications such as physics, engineering, and economics.When solving quadratic equations, one can use several methods:

• Factoring the equation.
• Completing the square.
• Using the quadratic formula.
• Graphical methods.

Each method is suitable for different situations depending on the equation's complexity and requirement. In our exercise, the equation \(x^2 - kx + k + 2\) was explored to find particular values of \(k\) that make the roots equal.

Quadratic Formula

The quadratic formula is a straightforward and universally applicable method to find the roots of any quadratic equation. Given the quadratic equation \(ax^2 + bx + c = 0\), its roots can be found using the formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The "\(\pm\)" symbol indicates that there are generally two solutions to consider: one that adds the square root and one that subtracts it. The flexibility of the quadratic formula makes it an essential tool since it can solve any quadratic equation.In the step-by-step solution provided, after setting the discriminant to zero, the quadratic formula was used to solve \(k^2 - 4k - 8 = 0\). It resulted in two potential values for \(k\): \(k = 6\) and \(k = -2\), where the original quadratic equation possesses equal roots. This illustration showcases the power and utility of the quadratic formula in determining precise solutions.