Problem 62 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(a\), the roots of the equation \(x^{2}+a^{2}=8 x+6 a\) are real.

Step-by-Step Solution

Answer

The roots of the given equation \(x^{2}+a^{2}=8 x+6 a\) are real for all values of \(a\).

Step 1: Rewrite the equation in the standard form

First, we need to rewrite the given equation \(x^{2}+a^{2}=8 x+6 a\) in the standard form: \(Ax^{2} + Bx + C = 0\). To do this, subtract \(8x\) and \(6a\) from both sides to get: \[x^{2} - 8x + a^{2} - 6a = 0\] Now we have the equation in the standard form with: \(A = 1\), \(B = -8\), and \(C = a^{2} - 6a\).

Step 2: Calculate the Discriminant

Now, we need to calculate the discriminant \(\Delta\). Remember that the formula for the discriminant is \(\Delta = b^{2} - 4ac\). In our case, A = 1, B = -8, and C = \(a^{2} - 6a\). Plug these values into the formula to get: \(\Delta = (-8)^{2} - 4(1)(a^{2} - 6a)\) \(\Delta = 64 - 4a^{2} + 24a\)

Step 3: Apply the condition for real roots

For the roots of the equation to be real, the discriminant must be non-negative (\(\Delta \geq 0\)). Therefore, we need to find the values of \(a\) for which: \(64 - 4a^{2} + 24a \geq 0\)

Step 4: Solve the inequality

To solve the inequality, rearrange the terms to get: \(4a^{2} - 24a + 64 \leq 0\) Divide both sides by 4 to simplify the inequality further: \(a^{2} - 6a + 16 \leq 0\) Now, to find the roots of the quadratic equation \(a^{2} - 6a + 16 = 0\), we can use the quadratic formula: \(a = \frac{-B \pm \sqrt{\Delta'}}{2A}\) In this case, \(A' = 1\), \(B' = -6\), and \(C' = 16\). Calculate the new discriminant \(\Delta'\), which is different from \(\Delta\) in Step 2: \(\Delta' = B'^{2} - 4A'C' = (-6)^{2} - 4(1)(16) = 36 - 64 = -28\) Since \(\Delta' < 0\), the equation \(a^{2} - 6a + 16 = 0\) has no real roots. This means that the inequality \(a^{2} - 6a + 16 \leq 0\) must be true for all values of \(a\), since the expression inside the inequality never reaches zero. Thus, the roots of the given equation \(x^{2}+a^{2}=8 x+6 a\) are real for all values of \(a\).

Key Concepts

Quadratic Equation

A quadratic equation is an equation in which the highest degree of the variable is two. It's generally written in the form:

• \( Ax^2 + Bx + C = 0 \)

This is known as the "standard form" of a quadratic equation. To recognize a quadratic equation, look for:

• A squared term (like \(x^2\))
• A linear term (like \(x\))
• A constant term (like \(C\))

In our exercise, we started with the equation \(x^{2}+a^{2}=8 x+6 a\) and transformed it into the standard form of the quadratic equation: \(x^2 - 8x + a^2 - 6a = 0\). Using the standard form makes it easier to solve the equation by using algebraic methods or formulas to find the roots.

Discriminant

The discriminant is a component of the quadratic formula used to determine the nature of the roots of a quadratic equation. The formula involves three coefficients \(A\), \(B\), and \(C\) from the standard form \(Ax^2 + Bx + C = 0\). The discriminant is given by:

• \( \Delta = B^2 - 4AC \)

The value of the discriminant tells us:

• If \( \Delta > 0 \): Two distinct real roots
• If \( \Delta = 0 \): One real root (also called a repeated root)
• If \( \Delta < 0 \): No real roots (the roots are complex or imaginary)

In our example, the discriminant was calculated as \( \Delta = 64 - 4a^2 + 24a \). To ensure the roots are real, the discriminant must be non-negative. This led us to form and solve an inequality.

Inequality

An inequality expresses a relationship where expressions are limited to being greater or less than each other, rather than simply equal. In quadratic equations, inequalities often determine the conditions under which solutions (or roots) exist.To solve the inequality, we ensure:

• The discriminant \( \Delta \geq 0 \)

From our exercise:

• The inequality formed was \( 64 - 4a^2 + 24a \geq 0 \)
• This was simplified to \( a^2 - 6a + 16 \leq 0 \)

However, upon further calculation, it was found that this inequality holds true for all real numbers. It indicates that the given quadratic equation has real roots for any value of \(a\). This practical use of inequalities aids in understanding and predicting the nature of solutions that a quadratic equation can have.

Standard Form

The "standard form" is a template that helps identify, manipulate, and solve quadratic equations systematically. This form is widely recognized as:

• \( Ax^2 + Bx + C = 0 \)

With this structure, specific tools like the quadratic formula or factoring can be easily applied. Each part – the squared term \(Ax^2\), the linear term \(Bx\), and the constant \(C\) – contributes to defining the equation's characteristics: the direction of its graph, its axis of symmetry, and its roots. In our example, transforming the equation into standard form as \(x^2 - 8x + a^2 - 6a = 0\) allowed us to directly identify \(A = 1\), \(B = -8\), and \(C = a^2 - 6a\). This identification is essential because it lays groundwork for further calculations like the discriminant, helping us understand the conditions needed for real roots.