Question

Hardik and Shivang attempted to solve a quadratic equation Hardik made a mistake in writing down the constant term and ended up in roots \((4,3)\) Shivang made a mistake in writting down coefficient of \(x\) to get roots \((3,2)\) The correct roots of equation are (a) \(-4,3\) (b) 6,1 (c) 4,3 (d) \(-6,-1\)

Short answer

The correct roots of the equation are (Option b) 6 and 1.

Step-by-step solution

Identifying the Incorrect Equations

Given, Hardik made a mistake in writing the constant term and ended up with roots (4,3). The correct equation has the form \(ax^2+bx+c=0\), and Hardik's incorrect equation has the form \(ax^2+bx+d=0\) with roots 4 and 3. Thus, Hardik's incorrect equation can be written as: \[a(x-4)(x-3)=0\] Shivang made a mistake in writing the coefficient of x and got roots (3,2). The correct equation has the form \(ax^2+bx+c=0\), and Shivang's incorrect equation has the form \(ax^2+kx+c=0\) with roots 3 and 2. Thus, Shivang's incorrect equation can be written as: \[a(x-3)(x-2)=0\]

Finding the correct constant term and coefficient of x

To find the correct constant term and coefficient of x, let's expand both equations. Hardik's incorrect equation: \[a(x-4)(x-3)=ax^2-7ax+12a=0\] Shivang's incorrect equation: \[a(x-3)(x-2)=ax^2-5ax+6a=0\] Comparing the coefficients, we can see that the coefficients of \(x^2\) are equal in both equations, so the correct equation will also have the coefficient of \(x^2\) as 'a.' Now, let's find the correct coefficient of x. Notice that the middle terms of the two equations differ by 2a (-7a and -5a). So, the correct coefficient of x must be either -5a (if Hardik made a mistake) or -7a (if Shivang made a mistake). Similarly, the constant term differs by 6a (12a and 6a). So, the correct constant term must be either 6a (if Hardik made a mistake) or 12a (if Shivang made a mistake).

Analyzing the Given Options

Now, let's analyze each option to determine which one is correct. (a) \(-4,3\) If this is the correct option, then the correct equation can be represented as: \[a(x+4)(x-3)=ax^2+ax-12a\] Here, we have a correct coefficient of x as -7a, but not the correct constant term as 12a. Therefore, this option is not correct. (b) 6,1 If this is the correct option, then the correct equation can be represented as: \[a(x-6)(x-1)=ax^2-7ax+6a\] Here, both coefficients of x and the constant term are correct. Therefore, this is the correct option. (c) 4,3 We don't need to check this option, as we have already found the correct answer. (d) \(-6,-1\) We don't need to check this option, as we have already found the correct answer. So, the correct roots of the equation are (Option b) 6 and 1.

Key concepts

Roots of Equations

In mathematics, finding the roots of an equation equates to determining the solutions for the equation. In the context of a quadratic equation, it takes the general form: \[ ax^2 + bx + c = 0 \] Here, the roots are the values of \( x \) that make the equation true. For quadratic equations, these roots can be found by factoring the equation or using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In the given exercise, we examined scenarios where incorrect constants or coefficients introduced errors. When Hardik made a mistake with the constant term, his polynomial generated roots \( 4 \) and \( 3 \). Meanwhile, Shivang altered the coefficient of \( x \), leading to roots \( 3 \) and \( 2 \). Each of these scenarios demonstrates how slight changes in the equation fundamentally impact the computed roots, thereby emphasizing precision in defining every component of the polynomial.

Coefficients in Polynomials

Coefficients in polynomials are multiplicative factors attached to the terms. In a quadratic equation, these could be indicative of slope or scaling. The general form includes:

• \( a \) - coefficient of \( x^2 \)
• \( b \) - coefficient of \( x \)
• \( c \) - constant term

In Shivang's scenario, the coefficient \( b \), was modified, giving different roots. By altering \( b \), he effectively changed the symmetry and position of the parabola (represented by the quadratic equation graph) along the x-axis. In mathematics, this coefficient not only influences the position but the direction a parabola opens, thus affecting what the roots or solutions of the equation will be. Incorrectly choosing \( b \) can quickly lead to different and sometimes nonsensical solutions in the context of a problem.

Expansion of Binomials

The expansion of binomials is a process by which expressions involving binomials are multiplied out into polynomial form. When we solve equations like the ones in the problem, starting from the factored form such as \((x-r_1)(x-r_2)=0\), where \( r_1 \) and \( r_2 \) are roots: \[ (x - 4)(x - 3) = x^2 - 7x + 12 \] This expansion demonstrates how numerically plugging in root values into an expression converts it into a fully realized quadratic polynomial. By blindly expanding, without concern for precision, erroneous constants and coefficients will emerge, affecting the integrity of the equation's solutions. Expanding binomials into polynomials requires attention to coefficient positioning and term math. It is vital to apply each term from one binomial to all terms in the other binomial, highlighting the distributive property role in these expansions.