Introduction to Perfect Square Quadratic Equations

In mathematics, a quadratic equation can be a perfect square if its roots are equal, which means the discriminant of the equation is zero. This article will explore the process of finding the value of k for which the given quadratic equation becomes a perfect square.

Understanding the Quadratic Equation and Its Components

Consider the quadratic equation:

$$kx^2 - (k-1)x - (k-1) 0.$$

This equation can be rewritten as:

$$ax^2 bx c 0,$$

where a k, b k-1, and c k-1. For this quadratic equation to be a perfect square, the discriminant D must be zero. The discriminant is given by:

$$D b^2 - 4ac.$$

Calculating the Discriminant

Substituting the values of a, b, and c, we get:

$$D (k-1)^2 - 4k(k-1).$$

First, we expand and simplify the expression:

$$(k-1)^2 - 4k(k-1) k^2 - 2k 1 - 4k^2 4k.$$

Combining like terms, we obtain:

$$D k^2 - 4k^2 - 2k 4k 1 -3k^2 2k 1.$$

Setting the discriminant equal to zero, we have:

$$-3k^2 2k 1 0.$$

For simplification, we multiply the equation by -1:

$$3k^2 - 2k - 1 0.$$

Solving the Equation Using the Quadratic Formula

Using the quadratic formula:

$$k frac{-b pm sqrt{b^2 - 4ac}}{2a},$$

where a 3, b -2, and c -1, we get:

$$D (-2)^2 - 4 cdot 3 cdot (-1) 4 12 16.$$

Substituting b, a, and c into the quadratic formula, we have:

$$k frac{--2 pm sqrt{16}}{2 cdot 3} frac{2 pm 4}{6}.$$

This gives us two possible solutions for k:

$$k frac{2 4}{6} frac{6}{6} 1$$ $$k frac{2 - 4}{6} frac{-2}{6} -frac{1}{3}.$$

Therefore, the values of k for which the quadratic equation becomes a perfect square are:

$$k 1 text{ and } k -frac{1}{3}.$$

Conclusion

This article has explained the process of determining the value of k for which the given quadratic equation becomes a perfect square by setting the discriminant to zero. The values of k are 1 and -frac{1}{3}. Understanding this concept is essential for solving various mathematical problems involving quadratic equations.