Problem 332 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Solve the equation \((a-1) x^{2}+2(2 a+1) x+(4 a+3)=0\).

Step-by-Step Solution

Answer

The solutions of the equation \((a-1) x^{2}+2(2 a+1) x+(4 a+3)=0\) in terms of \(a\) can be found by applying the quadratic formula and simplifying the resulting expression.

Step 1: Identify the coefficients

From the given quadratic equation, we can equate it to the general form \( Ax^2 + Bx + C = 0 \). Thus, \( A = (a-1) \), \(B = 2(2a+1)\) and \(C = (4a+3)\).

Step 2: Apply the quadratic formula

Substitute \(A\), \(B\), and \(C\) into the quadratic formula: \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\)

Step 3: Simplify the solutions

Simplify the above expression to get the solutions of \(x\) in terms of \(a\) by performing arithmetic operations.

Key Concepts

Coefficients Identification

When faced with a quadratic equation, the first step is to identify the coefficients. These coefficients are part of the expression that defines the equation in its standard quadratic form. A quadratic equation looks like this:
\[ Ax^2 + Bx + C = 0 \]
This is a universal format where \(A\), \(B\), and \(C\) represent constants, known as coefficients.

For the given problem, the quadratic equation
\((a-1) x^{2}+2(2 a+1) x+(4 a+3)=0\)
needs to be matched to the general form. We identify:

• \(A = (a-1)\), which is the coefficient of \(x^2\)
• \(B = 2(2a+1)\), which is the coefficient of \(x\)
• \(C = 4a+3\), which is the constant term

Recognizing these coefficients is essential. It helps in substituting them into the quadratic formula for further calculations.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. Once you have identified the coefficients \(A\), \(B\), and \(C\), you can plug them into this formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
This formula provides two solutions for \(x\), depending on whether you add or subtract the square root part.

For the given equation, substitute:

• \(A = (a-1)\)
• \(B = 2(2a+1)\)
• \(C = 4a+3\)

Inserting these into the formula will allow you to work through the calculations to find the possible values of \(x\). This method is reliable since it can solve any quadratic equation, whether it has real or complex roots.

Simplification of Solutions

After substituting the values into the quadratic formula and solving for \(x\), the work isn't complete yet. The expression you obtain often looks complicated. Simplifying this expression is a crucial step to make the solutions more understandable.

You will perform operations like:

• Calculating the determinant \(B^2 - 4AC\)
• Simplifying the square root \(\sqrt{B^2 - 4AC}\)
• Dividing through by \(2A\)

These steps involve arithmetic operations such as squaring, addition, and division. Each simplification helps in providing more concise and clearer solutions for \(x\) in terms of the variable \(a\).

Understanding solution simplification allows you not only to solve current problems but also to grasp the essence of quadratic equations more thoroughly.