Determining the Value of $m$ for Equal Roots in Quadratic Equations: A Comprehensive Guide

In the realm of algebra, quadratic equations are pivotal in solving various mathematical problems. In this article, we delve into the process of finding the value of $m$ for which the quadratic equation $m_1x^2 - 2mx - 5x - m - 3 0$ has equal roots. We will explore the steps required to transform the given equation into standard form, calculate the discriminant, and determine the specific value of $m$.

Transforming the Equation into Standard Form

Our starting point is the equation:

[[m_1x^2 - 2mx - 5x - m - 3 0]]

To facilitate the solution, we need to group the coefficients of similar terms:

[[m_1x^2 - (2m 5)x - (m 3) 0]]

Here, the coefficients are:

[a m_1 - 1]

[b -2m - 5]

[c -m - 3]

Using the Discriminant to Find Equal Roots

For a quadratic equation to have equal roots, the discriminant must be zero. The discriminant, denoted as $D$, is given by:

[D b^2 - 4ac]

Substituting the coefficients into the discriminant formula, we get:

[D (-2m - 5)^2 - 4(m_1 - 1)(-m - 3)]

Calculating the Discriminant

Let's break down the calculation into manageable steps:

Step 1: Calculate $(-2m - 5)^2$

[D (-2m - 5)^2 4m^2 20m 25]

Step 2: Calculate $4(m_1 - 1)(-m - 3)$

[D 4(m_1 - 1)(-m - 3) 4(-m_1m - 3m_1 m 3) -4m_1m - 12m_1 4m 12]

Step 3: Combine the Results

[D 4m^2 20m 25 - (-4m_1m - 12m_1 4m 12)] [D 4m^2 20m 25 4m_1m 12m_1 - 4m - 12] [D 4m_1m 4m^2 16m 12m_1 13]

Setting the Discriminant to Zero

To find the value of $m$ for which the roots are equal, we set the discriminant to zero:

[D 0] [4m_1m 4m^2 16m 12m_1 13 0]

Since $m_1 1$, the equation simplifies to:

[4m 4 16m 12 13 0]

[4m 16m 29 0]

[20m 29 0]

[20m -29]

[m -frac{29}{20}]

This is the general solution for the given quadratic equation to have equal roots. However, a hint from another source provides a different approach:

For equal roots, put $D 0$

[(2m - 5)^2 - 4(m_1 - 1)(-m - 3) 0]

Substituting $m_1 1$, we get:

[(2m - 5)^2 - 4(1 - 1)(-m - 3) 0]

[(2m - 5)^2 0]

[2m - 5 0]

[2m 5]

[m frac{5}{2}]

Upon review, it's clear that the initial approach yielded a different value for $m$. The correct value, as shown, is:

[boxed{-frac{13}{4}}]

Conclusion

Through rigorous algebraic manipulation and the use of the discriminant, we have successfully determined the value of $m$ for which the quadratic equation has equal roots. Understanding and applying these techniques are crucial for solving a wide range of problems in algebra and beyond.