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

Question

If equations \(a x^{2}+b x+c=0\) and \(c x^{2}+b x\) \(+a=0\) have one root common, show that either \(a+b+c=0\) or \(a-b+c=0\).

Step-by-Step Solution

Answer

The common root implies either \(a+b+c=0\) or \(a-b+c=0\).

Step 1: Understand the Problem

We are given two quadratic equations: \(a x^{2}+b x+c=0\) and \(c x^{2}+b x+a=0\). If they have one root in common, we need to show that either \(a+b+c=0\) or \(a-b+c=0\) must be true.

Step 2: Consider the Common Root

Let \(\alpha\) be the common root. Then it satisfies both equations: \(a \alpha^{2}+b \alpha+c=0\) and \(c \alpha^{2}+b \alpha+a=0\).

Step 3: Subtract the Equations

Consider \(a \alpha^{2}+b \alpha+c=0\) and \(c \alpha^{2}+b \alpha+a=0\). Subtract the first equation from the second to eliminate \(b \alpha\): \((c - a)\alpha^2 + (a - c) = 0\).

Step 4: Factorize the Resulting Equation

The subtracted equation can be rewritten as \((c - a)(\alpha^2 - 1) = 0\). This implies either \(c - a = 0\) or \(\alpha^2 - 1 = 0\), leading to \(\alpha = \pm 1\).

Step 5: Set Conditions for Solutions

If \(c-a=0\), then \(c = a\). If \(\alpha = 1\), substitute in the original equations to get \(a + b + c = 0\). If \(\alpha = -1\), substitute to obtain \(a - b + c = 0\).

Step 6: Conclusion

From the conditions found, if the common root \(\alpha\) is \(1\) or \(-1\), we have \(a + b + c = 0\) or \(a - b + c = 0\) respectively.

Key Concepts

Common Roots

Understanding common roots in the context of quadratic equations is crucial. A common root is a value that satisfies both of the given algebraic equations. It means that when you substitute this root back into the equations, both sides are equal. When two quadratic equations share a common root, their solutions intersect at this point.

In our exercise, we have two quadratic equations, each of the form \(ax^2 + bx + c = 0\). We need to find conditions under which they possess a common root, \(\alpha\). This common root satisfies both \(a\alpha^2 + b\alpha + c = 0\) and \(c\alpha^2 + b\alpha + a = 0\).

By working through the equations, we can simplify and manipulate them to identify relationships between their coefficients \(a\), \(b\), and \(c\) based on the common root's value.

Algebraic Equations

Algebraic equations are mathematical statements that involve variables and constants arranged through operations like addition, multiplication, etc. These equations help us describe relationships and solve for unknowns. Quadratic equations are a special type of algebraic equation where the highest power of the variable is two. They generally take the form \(ax^2 + bx + c = 0\).

The problem provides two such quadratic equations. The solution involves showing how specific algebraic manipulations can lead to identifying conditions (like \(a+b+c = 0\) or \(a-b+c = 0\)) that must be met based on particular algebraic properties of these equations. This involves using properties like symmetry between terms and zero-product reasoning, which are unique to quadratic algebraic equations.

Understanding how algebraic solutions lead to logical deductions is the key to dynamic problem-solving and showcases the power of formulas within algebra.

Polynomial Roots

Polynomial roots are solutions to polynomial equations. A root is a value such that when substituted into the polynomial equation, it yields zero. For a quadratic polynomial such as \(ax^2 + bx + c = 0\), the roots can be found using the quadratic formula, factoring, or completing the square.

For our exercise, we focus on a situation where the roots of two polynomials, or quadratic equations, overlap, making them common. In particular, because these equations have a common root, say \(\alpha\), it means:

• \(a\alpha^2 + b\alpha + c = 0\)
• \(c\alpha^2 + b\alpha + a = 0\)

Analyzing this, we discover particular values of \(\alpha\). These values help deduce the equations \(a + b + c = 0\) and \(a - b + c = 0\).

Through algebraic subtraction and solving, we identify these polynomial roots' properties which play a crucial role in understanding how polynomials intersect through their roots.