Problem 74 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Examine the nature of the roots of the quadratic \((b-x)^{2}-4(a-x)(c-x)=0\) where \(a, b, c\) are real.

Step-by-Step Solution

Answer

In the given quadratic equation \((b-x)^{2}-4(a-x)(c-x)=0\), the discriminant is given by \(\Delta = (2b - 4a - 4c)^2 - 4(-3)(4ad - 4ac + b^2 + 4bd)\). However, due to the complexity of the discriminant expression, providing general conditions on \(a, b, c\) for the nature of the roots is not feasible. To investigate the roots' nature for specific cases, we can substitute specific values of \(a, b, c\) and analyze the resulting discriminant value to determine if the roots are real and distinct, real and equal, or complex.

Step 1: Write the quadratic equation in standard form

We begin by expressing the given quadratic equation in the standard quadratic form, i.e., \(Ax^2 + Bx + C = 0\). This will help us in calculating the discriminant easily. Given, \((b-x)^{2}-4(a-x)(c-x)=0\)

Step 2: Expand and simplify the equation

Now, let's expand and simplify the equation: \((b^2 -2bx + x^2) - 4(ac - ad + cx - xd + ax - bd + x^2 - xd) = 0\)

Step 3: Collect terms and rewrite the equation

Group the terms with similar powers of x: \(x^2 - 2bx + b^2 - 4ac + 4ad - 4cx + 4xd - 4ax + 4bd - 4x^2 + 4xd = 0\) Combine the like terms: \(-3x^2 + 4ad - 4ac + 2bx + b^2 + 4bd - 4ax - 4cx = 0\)

Step 4: Calculate the discriminant

The discriminant of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by \(\Delta = B^2 - 4AC\). From the equation in step 3, we have: \(A = -3\) \(B = 2b - 4a - 4c\) \(C = 4ad - 4ac + b^2 + 4bd\) Now, calculate the discriminant \(\Delta\): \(\Delta = (2b - 4a - 4c)^2 - 4(-3)(4ad - 4ac + b^2 + 4bd)\)

Step 5: Simplify and analyze the discriminant

As we know, the nature of the roots can be determined by the discriminant value: - If \(\Delta > 0\), roots are real and distinct. - If \(\Delta = 0\), roots are real and equal. - If \(\Delta < 0\), roots are complex. The task is to examine the nature of the roots given that \(a, b, c\) are real numbers. For this case, we need to find the conditions on \(a, b, c\) that will make \(\Delta\) positive, negative, or zero. However, due to the complexity of the discriminant expression, it is not possible to provide a straightforward analysis or general conditions for the roots' nature for all real numbers \(a, b, c\). An approach to analyze the roots' nature would be to solve the discriminant expression for specific values of \(a, b, c\) and observe the conditions leading to different roots' natures.

Key Concepts

Discriminant Analysis

The discriminant plays a crucial role in understanding the nature of the roots of a quadratic equation. It is derived from the standard form of the quadratic equation, \(Ax^2 + Bx + C = 0\), and is defined as \(\Delta = B^2 - 4AC\). By calculating this value, we can determine how the roots will behave.

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), the equation has exactly one real root (a repeated root).
• If \(\Delta < 0\), the equation has two complex roots.

In the solution provided, the discriminant \(\Delta\) is evaluated for a more complex quadratic originating from the expression \((b-x)^{2} - 4(a-x)(c-x) = 0\). This leads to the expression \[-3x^2 + 2bx + b^2 + 4ad - 4ac + 4bd - 4ax - 4cx = 0\].
From here, the calculation of \(\Delta\) involves identifying the coefficients (\(A = -3\), \(B = 2b - 4a - 4c\), \(C = 4ad - 4ac + b^2 + 4bd\)) and substituting them into the formula, providing insight into the potential nature of the roots for different real numbers \(a, b, c\).

Nature of Roots

The nature of the roots of a quadratic equation depends on the value of the discriminant \(\Delta\), which reveals what kind of solutions can be expected.
### Real RootsReal roots occur when \(\Delta \geq 0\).

• If \(\Delta = 0\), the roots are real and repeated, meaning the parabola touches the x-axis at only one point.
• If \(\Delta > 0\), the roots are real and distinct, indicating that the parabola intersects the x-axis at two different points.

### Complex RootsWhen \(\Delta < 0\), the quadratic equation has complex roots. These are not points on the x-axis of the graph, but rather they represent intersections in the complex plane.
Exploring the nature of roots for the equation \((b-x)^2 - 4(a-x)(c-x) = 0\) requires examining \(\Delta\) for specific cases of \(a, b, c\). This complexity often necessitates technological assistance or special cases to solve definitively.

Real and Complex Numbers

Understanding real and complex numbers is essential when analyzing quadratic equations. Real numbers include all the typical numbers we encounter in daily life: positive numbers, negative numbers, and zero. These are the numbers that appear on the standard number line.
However, not all quadratic equations have solutions among real numbers alone. When the discriminant \(\Delta\) is negative, the roots are complex. Complex numbers are numbers that include the imaginary unit \(i\), where \(i^2 = -1\). They are typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

• Real Numbers: Solutions appear on the x-axis (\(\Delta \geq 0\)).
• Complex Numbers: Solutions extend into the complex plane (\(\Delta < 0\)).

The quadratic equation derived from \((b-x)^2 - 4(a-x)(c-x) = 0\) demands assessing whether the results are purely real or if they venture into the complex realm, depending on the values of \(a\), \(b\), and \(c\). Understanding these fundamentals supports a deeper comprehension of any quadratic equation's results.