Question

Rohan ond Sohan were attempting to solve the quadratic equation, \(\mathrm{x}^{2}-\mathrm{ax}+\mathrm{b}=0 .\) Rohan copied the coefficient of \(x\) wrongly and obtained the roots as 4 and \(12 .\) Sohan copied the constant term wrongly and obtained the roots as \(-19\) and 3 . Find the correct roots. (1) \(-2\) and \(-24\) (2) 2 and 24 (3) 4 and 12 (4) \(-4\) and \(-12\)

Short answer

Answer: The correct roots are 19 and -3.

Step-by-step solution

Sum of roots given by Rohan

We are given that the roots obtained by Rohan are 4 and 12. The sum of these roots should equal the correct sum of the roots of the quadratic equation, so we have: Roots Sum = 4 + 12 = 16 Now, we know that the sum of roots of a quadratic equation \(x^2 - ax + b = 0\) is equal to \(\dfrac{-(-a)}{1} = a\). Therefore, we get: \(a = 16\)

Product of roots given by Sohan

We are given that the roots obtained by Sohan are -19 and 3. The product of these roots should equal the correct product of the roots of the quadratic equation, so we have: Roots Product = (-19) * (3) = -57 Now, we know that the product of roots of a quadratic equation \(x^2 - ax + b = 0\) is equal to \(\dfrac{b}{1} = b\). Therefore, we get: \(b = -57\)

The correct quadratic equation

Now that we have found the correct values for 'a' and 'b', the correct quadratic equation is: \(x^2 - 16x - 57 = 0\)

Find the correct roots using Quadratic Formula

We can find the correct roots using the Quadratic Formula: \(\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\). In this case, \(a=1, b=-16,\) and \(c=-57\). Plugging these values into the formula, we get: \(x = \dfrac{-(-16)\pm\sqrt{(-16)^2-4(1)(-57)}}{2(1)}\) \(x = \dfrac{16\pm\sqrt{256 + 228}}{2}\) \(x = \dfrac{16\pm\sqrt{484}}{2}\) \(x = \dfrac{16\pm 22}{2}\) Now, we can find the two roots using the two different results from the \(\pm\) part of the formula: \(x_1 = \dfrac{16+22}{2} = \dfrac{38}{2} = 19\) \(x_2 = \dfrac{16-22}{2} = \dfrac{-6}{2} = -3\) So the correct roots are 19 and -3, which are not in the given options.

Key concepts

Roots of Quadratic Equations

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the solutions for \(x\) are deemed the 'roots' of the equation. These roots can be either real or complex numbers.
Understanding how to manipulate and solve for these roots is fundamental to solving any quadratic equation. The roots tell us where the graph of the quadratic polynomial intersects the x-axis.
For example, if the quadratic equation is \(x^2 - ax + b = 0\), the correct roots are the values of \(x\) which satisfy the equation and make it true. Identifying correct coefficients helps in finding accurate roots.

Quadratic Formula

To find the roots of any quadratic equation \(ax^2 + bx + c = 0\), we use the Quadratic Formula:
\[\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\]
This formula will always provide the roots, whether they are real or complex. The expression under the square root, known as the discriminant, \(b^2 - 4ac\), determines the nature of the roots.

• If the discriminant is positive, we have two distinct real roots.
• If it equals zero, the roots are real and identical.
• If it's negative, the roots are complex and conjugate.

In the given problem, we use the formula to find the correct roots, resulting in \(x = 19\) and \(x = -3\).

Sum of Roots

For the quadratic equation \(ax^2 + bx + c = 0\), the sum of its roots, given by \(x_1 + x_2\), is derived from the equation as:
\[x_1 + x_2 = -\frac{b}{a}\]
This relationship comes from a fundamental property of quadratic equations and is useful for deducing unknown coefficients if roots are known.
In the exercise, even when Rohan made an error in the coefficient of \(x\), the sum of roots, \(4 + 12 = 16\), led to finding \(a = 16\), affirming the root's relationship with the coefficients.

Product of Roots

Another important relation in quadratic equations is the product of the roots, \(x_1 \times x_2\), represented as:
\[x_1 \times x_2 = \frac{c}{a}\]
This arises from setting the equation to zero and represents the product of the values of \(x\) that satisfy the equation.
In our problem, Sohan deducing \(x_1 \times x_2 = (-19) \times 3 = -57\) allowed them to confirm \(c = -57\). This maintains consistency and accuracy reflecting the relationship between the roots and quadratic equation's coefficients.