Question

Find the roots of the equation \(\ell^{2}\left(\mathrm{~m}^{2}-\mathrm{n}^{2}\right) \mathrm{x}^{2}+\mathrm{m}^{2}\left(\mathrm{n}^{2}-\ell^{2}\right) \mathrm{x}+\mathrm{n}^{2}\left(\ell^{2}-\mathrm{m}^{2}\right)=0\) (1) \(1, \frac{\mathrm{n}^{2}\left(\ell^{2}-\mathrm{m}^{2}\right)}{\ell^{2}\left(\mathrm{~m}^{2}-\mathrm{n}^{2}\right)}\) (2) \(1, \frac{-\mathrm{m}^{2}\left(\ell^{2}-\mathrm{n}^{2}\right)}{\ell^{2}\left(\mathrm{~m}^{2}-\mathrm{n}^{2}\right)}\) (3) \(1, \frac{\mathrm{n}^{2}\left(\ell^{2}+\mathrm{m}^{2}\right)}{\ell^{2}\left(\mathrm{~m}^{2}-\mathrm{n}^{2}\right)}\) (4) \(1, \frac{-m^{2}\left(\ell^{2}+n^{2}\right)}{\ell^{2}\left(m^{2}-n^{2}\right)}\)

Short answer

Answer: \(x = \frac{-(n^{2}-\ell^{2}) \pm \sqrt{(n^{2} -2\ell^2 + \ell^2)(n^2 - \ell^2)}}{2\ell^{2}}\)

Step-by-step solution

Identify the coefficients

The given quadratic equation is \(\ell^{2}(m^{2}-n^{2})x^{2} + m^{2}(n^{2}-\ell^{2})x + n^{2}(\ell^{2}-m^{2}) = 0\) Notice that the coefficients are: - a = \(\ell^{2}(m^{2}-n^{2})\) - b = \(m^{2}(n^{2}-\ell^{2})\) - c = \(n^{2}(\ell^{2}-m^{2})\)

Apply the quadratic formula

We can now use the quadratic formula to find the roots: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Substitute coefficients into the quadratic formula

Now we will substitute the coefficients a, b, and c into the quadratic formula: \(x = \frac{-(m^{2}(n^{2}-\ell^{2})) \pm \sqrt{(m^{2}(n^{2}-\ell^{2}))^2 - 4(\ell^{2}(m^{2}-n^{2}))(n^{2}(\ell^{2}-m^{2}))}}{2(\ell^{2}(m^{2}-n^{2}))}\)

Simplify the expression

The given options suggest that we only need to focus on the expression inside the square root. Let's simplify that term in the discriminant: \(D = b^2 -4ac = (m^{2}(n^{2}-\ell^{2}))^2 - 4(\ell^{2}(m^{2}-n^{2}))(n^{2}(\ell^{2}-m^{2}))\) We can factor the discriminant as follows: \(D = [(m^2)(n^2 -\ell^2)]^2 - (4\ell^2 m^2 n^2)\) Notice that the terms \([(m^2)(n^2 -\ell^2)]^2\) and \((4\ell^2 m^2 n^2)\) have a common factor of \(m^2\). So to simplify the expression in the numerator, we can just cancel this common factor, resulting in the following expression: \(x = \frac{-(n^{2}-\ell^{2}) \pm \sqrt{((n^2 -\ell^2))^2 - 4\ell^2 n^2}}{2\ell^{2}} = \frac{-(n^{2}-\ell^{2}) \pm \sqrt{(n^{2} -2\ell^2 + \ell^2)(n^2 - \ell^2)}}{2\ell^{2}}\) Comparing this expression with the given options, we can find the correct roots. The correct answer is (2) $1, \frac{-\mathrm{m}^{2}\left(\ell^{2}-\mathrm{n}^{2}\right)}{\ell^{2}\left(\mathrm{~m}^{2}-\mathrm{n}^{2}\right)}$

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form ax² + bx + c = 0. The formula is given by

\[\begin{equation}x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\end{equation}\]

This formula provides the roots directly by substituting the coefficients, denoted by 'a', 'b', and 'c', from the equation. The part under the square root, b² - 4ac, is known as the discriminant, which tells us the nature of the roots. Using the quadratic formula simplifies the process of finding roots from factoring or completing the square. If the discriminant is positive, there are two distinct real roots; if zero, there is one real root; and if negative, there are no real roots in the real number system.

Discriminant

The discriminant in quadratic equations is a key determinant that reveals the number and type of roots without actually solving the equation. It is the expression b² - 4ac from the quadratic formula. In the problem we're discussing:

\[\begin{equation}D = b^2 - 4ac\end{equation}\]

Discriminant 'D' has been calculated using the given coefficients. Through simplification, the values of 'a', 'b', and 'c' are substituted, and the discriminant is factored to determine its value. With a positive discriminant, as in this case, we're guaranteed to find two distinct roots.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials that, when multiplied together, yield the original polynomial. In the context of quadratic equations, it often simplifies finding roots. For instance, if we had the quadratic equation ax² + bx + c, and we are able to factor it into (dx + e)(fx + g) = 0, we can easily find that the solutions are x = -e/d and x = -g/f.

When dealing with complex coefficients, as in the given problem, factoring by grouping and other advanced techniques may be necessary. In these cases, the quadratic formula can be a more straightforward path to the solution, especially when recognizing the role of the discriminant in simplification. Nevertheless, understanding factorization is crucial for recognizing patterns and simplifying expressions, leading to correct solutions.