Question

The condition for the equations \(a_{1} t^{2}+b_{1} t+c_{1}=0\) and \(a_{2} t^{2}+b_{2} t+c_{2}=0\) to have a common root is (a) \(a_{1} c_{2}-a_{2} c_{1}=b_{1} b_{2}\) (b) \(a_{1} a_{2}=b_{1} b_{2}-c_{1} c_{2}\) (c) \(\left(c_{1} a_{2}-c_{2} a_{1}\right)^{2}=\left(b_{1} c_{2}-b_{2} c_{1}\right)\left(a_{1} b_{2}-a_{2} b_{1}\right)\) (d) \(\left(b_{1} c_{2}-b_{2} c_{1}\right)^{2}=2\left(a_{1} b_{2}+a_{2} b_{1}\right)\)

Short answer

Based on the step by step solution provided, the condition for two quadratic equations \(a_{1} t^2 + b_{1} t + c_{1} = 0\) and \(a_{2} t^2 + b_{2} t + c_{2} = 0\) to have a common root is given by: \(\left(c_{1} a_{2}-c_{2} a_{1}\right)^{2}=\left(b_{1} c_{2}-b_{2} c_{1}\right)\left(a_{1} b_{2}-a_{2} b_{1}\right)\).

Step-by-step solution

Understand the concept of common root

A common root means that there exists a value of \(t\) that satisfies both equations simultaneously, i.e., makes both equations equal to zero. Let's denote this common root by \(t_0\). Then, both equations can be written as: 1. \(a_{1} t_0^{2}+b_{1} t_0+c_{1}=0\) 2. \(a_{2} t_0^{2}+b_{2} t_0+c_{2}=0\)

Eliminate t_0 from the equations

To find the condition, we need to eliminate \(t_0\) from the equations: We can multiply Equation 1 by \(a_2\) and Equation 2 by \(a_1\) and then subtract them to eliminate the \(t_0^2\) term: 1. \(a_{1}a_{2} t_0^{2}+(b_{1}a_{2}) t_0+(c_{1}a_{2})=0\) 2. \(a_{1}a_{2} t_0^{2}+(b_{2}a_{1}) t_0+(c_{2}a_{1})=0\) Subtracting Equation 2 from Equation 1, we get: \((b_{1}a_{2}-b_{2}a_{1})t_0 = (c_{1}a_{2}-c_{2}a_{1})\)

Rearrange the equation to find the condition

We need a condition that doesn't involve \(t_0\). Let's assume that t0 is non-zero, which allows us to divide both sides of the equation by \(t_0\) without changing the equality: \(\frac{(b_{1}a_{2}-b_{2}a_{1})t_0}{t_0} = \frac{(c_{1}a_{2}-c_{2}a_{1})}{t_0}\) This equation simplifies to: \((b_{1}a_{2}-b_{2}a_{1}) = \frac{(c_{1}a_{2}-c_{2}a_{1})}{t_0}\) Now we need to square both sides of the equation to eliminate the \(t_0\) term on the right side: \((b_{1}a_{2}-b_{2}a_{1})^2 = (c_{1}a_{2}-c_{2}a_{1})^2\)

Check which option matches our derived condition

Comparing the obtained equation with the given options, we can see that our condition matches option (c): \(\left(c_{1} a_{2}-c_{2} a_{1}\right)^{2}=\left(b_{1} c_{2}-b_{2}c_{1}\right)\left(a_{1} b_{2}-a_{2} b_{1}\right)\) So, the correct answer is (c).

Key concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( x \) is the variable. These equations have a characteristic quality of having a degree of two, meaning the highest power of the variable is two. Each quadratic equation can have up to two solutions, known as roots. These roots can be real or complex numbers.

To solve a quadratic equation, one can use various methods, such as:

• Factoring: Expressing the quadratic as a product of two binomials.
• Completing the square: Transforming the equation into a perfect square trinomial.
• Quadratic formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots.

In problems where two quadratic equations share a common root, special conditions are used to determine which value of \( x \) is a root for both equations simultaneously. These conditions can be derived through algebraic manipulation, as seen in the step-by-step solution provided for the given exercise.

Algebraic Manipulation

Algebraic manipulation involves the re-arranging and transforming equations to make them easier to solve or compare. When dealing with quadratic equations, this often includes operations such as addition, subtraction, multiplication, and division to eliminate unwanted variables or to isolate specific terms.

In the context of finding a common root for two quadratics, algebraic manipulation is used to eliminate the variable \( t_0 \) from the equations.
This is done by strategically multiplying the equations by coefficients and then subtracting one from the other. In this exercise, the steps involved multiplying one equation by the coefficient of the other equation and vice versa to eliminate the squared term. By doing so, the equations were reduced to a simpler form where \( t_0 \) could be cancelled out.

The final result is a condition that does not involve \( t_0 \), creating a direct comparison between the coefficients of the quadratics, which reveals under what condition these quadratics will share a common root.

Simultaneous Equations

Simultaneous equations refer to a set of equations with multiple variables that are solved together. The main goal is to find one set of values for the variables that satisfies all the given equations simultaneously.

When considering quadratic equations with a common root, the approach of simultaneous equations can reveal crucial insights. By treating the equations as simultaneous, you can apply strategies to solve for the shared value of the variable, \( t_0 \), that makes both equations zero.

For example:

• You would first write down the equations both with the unknown root \( t_0 \).
• Next, use algebraic methods such as elimination to remove \( t_0 \) from these equations.
• Finally, simplify to find a condition involving only the coefficients of the quadratics.

The result is a condition or a criterion that these quadratics must meet to have that common root value. Utilizing the principles of simultaneous equations aids in clarifying whether or not the initial conditions proposed in the exercise are satisfied by the derived condition.