Problem 157 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the equations \(x^{2}+2 x+3=0 \quad\) and \(a x^{2}+b x+c=0, a, b, c \in R\) have a common root, then \(a: b: c\) is (A) \(3: 2: 1\) (B) \(1: \underline{3: 2}\) (C) \(3: 1: 2\) (D) \(1: 2: 3\)

Step-by-Step Solution

Answer

The ratio \(a:b:c\) is \(1:2:3\).

Step 1: Identify the Roots of the Quadratic Equation

To find the roots of the equation \(x^2 + 2x + 3 = 0\), we need to use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = 2\), and \(c = 3\).

Step 2: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) of the quadratic equation. Substitute \(b = 2\), \(a = 1\), and \(c = 3\): \[ b^2 - 4ac = 2^2 - 4 \times 1 \times 3 = 4 - 12 = -8. \] Since the discriminant is negative, the roots of the equation \(x^2 + 2x + 3 = 0\) are complex and conjugate.

Step 3: Determine the Roots

Using the discriminant, the roots of the equation \(x^2 + 2x + 3 = 0\) are:\[ x = \frac{-2 \pm \sqrt{-8}}{2} = -1 \pm i\sqrt{2} \] Thus, the roots are complex conjugates \(-1 + i\sqrt{2}\) and \(-1 - i\sqrt{2}\). Let one of these roots, say \(\alpha = -1 + i\sqrt{2}\), be the common root.

Step 4: Substitute the Common Root in the Second Equation

Since \(\alpha\) is a root of both equations, it must satisfy the equation \(a x^2 + b x + c = 0\). Substitute \(x = -1 + i\sqrt{2}\) into the equation:\[a(-1+i\sqrt{2})^2 + b(-1+i\sqrt{2}) + c = 0.\]

Step 5: Expand \((-1 + i\sqrt{2})^2\)

Calculate \((-1 + i\sqrt{2})^2\):\[(-1 + i\sqrt{2})^2 = 1 - 2 \cdot i\sqrt{2} - 2 = -1 - 2i\sqrt{2}.\]

Step 6: Form the New Equation

Substitute the expanded form:\[a(-1 - 2i\sqrt{2}) + b(-1 + i\sqrt{2}) + c = 0.\]

Step 7: Set Up a System of Equations

For the imaginary part to be zero, the coefficients of \(i\sqrt{2}\) and constant terms must equal zero separately:- Imaginary part: \[-2a + b = 0\] - Real part: \[-a - b + c = 0\]

Step 8: Solve the System of Equations

Solve \(-2a + b = 0\) to obtain \(b = 2a\). Substitute \(b = 2a\) into \(-a - b + c = 0\):\[-a - 2a + c = 0 \Rightarrow -3a + c = 0 \Rightarrow c = 3a.\]

Step 9: Determine the Ratios of a, b, c

With \(b = 2a\) and \(c = 3a\), the ratio \(a:b:c\) is \(a:2a:3a\), which simplifies to \(1:2:3\).

Key Concepts

Complex Roots

Complex roots arise in quadratic equations when the discriminant is negative. In the given quadratic equation \(x^2 + 2x + 3 = 0\), calculating the discriminant \(b^2 - 4ac\) results in \(-8\), indicating complex roots. These roots cannot be expressed as real numbers and involve imaginary numbers, which are multiples of \(i\), the imaginary unit. The complex roots are conjugates, meaning they have the form \(a + bi\) and \(a - bi\), where \(a\) and \(b\) are real numbers.
For the given equation, the roots are \(-1 + i\sqrt{2}\) and \(-1 - i\sqrt{2}\). These roots reflect the symmetry in the complex plane, positioned equidistantly from the real axis.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides the roots through the expression:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Using this formula helps find roots regardless of whether they are real or complex. For our equation \(x^2 + 2x + 3 = 0\), substituting \(a = 1\), \(b = 2\), and \(c = 3\) into the formula gives us complex roots due to the negative discriminant.
The quadratic formula is universally applicable, providing a seamless way to discover the nature of the roots through the discriminant within the formula itself.

Discriminant Analysis

Discriminant analysis involves evaluating the expression \(b^2 - 4ac\) to determine the nature of the roots of a quadratic equation. It reflects the geometry of the equation’s solutions:

• If positive, the equation has two distinct real roots.
• If zero, it has exactly one real root, also known as a repeated root.
• If negative, the equation has complex conjugate roots.

In the exercise, calculating the discriminant for the equation \(x^2 + 2x + 3 = 0\) resulted in \(-8\). This negative value signals the presence of complex roots, driving us to solve using the quadratic formula. Understanding discriminant analysis is crucial because it informs the nature and quantity (real or complex) of the solutions before solving the equation fully.