Problem 14 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The number of values of \(a\) for which \(\left(a^{2}-3 a+2\right) x^{2}+\left(a^{2}-5 a+6\right) x+a^{2}-4=0\) is an identity in \(x\), is (a) 0 (b) 2 (c) 1 (d) 3

Step-by-Step Solution

Answer

The number of values of \(a\) is 1.

Step 1: Understand the Condition for Identity

For the given quadratic to be an identity in \(x\), the expression \((a^{2}-3a+2)x^{2}+(a^{2}-5a+6)x+a^{2}-4=0\) must equal zero for all values of \(x\). This occurs if each coefficient of \(x\) is zero, since this would make the whole expression zero regardless of \(x\)'s value.

Step 2: Set the Coefficients to Zero

The condition for identity requires each part of the quadratic to be zero:1. \(a^{2}-3a+2 = 0\)2. \(a^{2}-5a+6 = 0\)3. \(a^{2}-4 = 0\)

Step 3: Solve the Equations

We solve each of the equations obtained in Step 2.1. Solve \(a^{2}-3a+2=0\):Using the quadratic formula \(a = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), we find the roots to be \(a = 1, 2\).2. Solve \(a^{2}-5a+6=0\):This gives roots \(a = 2, 3\).3. Solve \(a^{2}-4=0\):This gives roots \(a = 2, -2\).

Step 4: Find Common Solutions

The values of \(a\) that make all three coefficients zero will satisfy the identity condition for this quadratic equation. We find the common roots across the solutions from step 3. The common value is \(a = 2\).

Step 5: Determine the Number of Values

Since \(a = 2\) is the only common solution that makes all three terms zero, there is only one such value. Thus, there is only 1 value of \(a\) that satisfies the condition of being an identity in \(x\).

Key Concepts

Identity Condition in Quadratic Equations

To understand when a quadratic equation is considered an identity in relation to a variable, let's dig deeper into this concept. A quadratic equation, \[ ax^2 + bx + c = 0 \] is an "identity" in the variable \(x\) when it holds true for all values of \(x\). This condition can only occur if every term in the equation individually equates to zero. This means:

• The coefficient of \(x^2\), represented as \(a\)
• The coefficient of \(x\), represented as \(b\)
• The constant term, represented as \(c\)

Each of these coefficients must be zero. When this occurs, the quadratic equation becomes \[ 0\cdot x^2 + 0\cdot x + 0 = 0 \] which simplistically translates to \(0 = 0\), a situation that is universally true for any \(x\). This is a key understanding needed to identify identity conditions: all parts must individually resolve to zero.
This principle applies to any standard quadratic form, ensuring that the equation remains universally unchanged across all possible values of its variable.

Solving Quadratic Equations

Solving quadratic equations, like \[ ax^2 + bx + c = 0 \] involves finding values of the variable that make the equation true. Often, we use the quadratic formula to find these solutions:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(b^2 - 4ac\) is known as the discriminant, and it tells us whether real solutions exist:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution, known as a "double root."
• If \(b^2 - 4ac < 0\), there are no real solutions; instead, we encounter complex solutions.

For our problem, alternate methods include factoring or completing the square, depending on the specific form of the quadratic equation. However, for understanding or identifying the roots where we set different quadratic components to zero, these traditional methods of solution are paramount.

Common Roots in Equations

Understanding common roots in quadratic equations is crucial when dealing with multiple equations that require a common solution. In other words, we're identifying a value of \(a\) or another variable that makes all given expressions simultaneously true. This requires us to determine the solutions or roots of each individual equation first and then find where these solutions overlap.

Let's consider three different quadratic expressions that are set to zero:

• \(a^2 - 3a + 2 = 0\)
• \(a^2 - 5a + 6 = 0\)
• \(a^2 - 4 = 0\)

The solutions to these equations, respectively, reveal potential roots. By identifying which solutions are shared among all equations, we find the common roots. In this example, solving each gives us solutions, and the number that can commonly be found across all is \(a = 2\). Finding common roots demands an acute awareness of how each equation relates through their solutions, highlighting a mutual solution or root across multiple equations.