Introduction to Quadratic Equations and Perfect Squares
Quadratic equations, expressions of the form ax^2 bx c 0, where a, b, and c are constants, are widely studied in mathematics. Among these, equations that can be transformed into perfect squares (of the form (px q)^2 0) exhibit special properties. In this article, we explore how to determine the value of k for which the given quadratic equation is a perfect square.
Determining the Conditions for a Perfect Square
A quadratic equation can be a perfect square if its discriminant (D) is equal to zero. The discriminant is given by the formula (D b^2 - 4ac). For a quadratic equation of the form (4 - kx^2 2k4x 8k1 0) to be a perfect square, its discriminant must be zero. Let's go through the steps to find the value of k that satisfies this condition.
Step 1: Setting up the Equation
The given equation is (4 - kx^2 2k4x 8k1 0). To find the value of k for which this equation is a perfect square, we equate the discriminant to zero:
#34;b^2 - 4ac 0#34;
Here, (a 4 - k), (b 2k4), and (c 8k1).
Step 2: Substituting the Values
Substituting these values into the discriminant formula, we get:
[ (2k4)^2 - 4(4 - k)(8k1) 0 ]
Simplifying this equation:
[ 4k4^2 - 4(4 - k)(8k1) 0 ]
Further simplification:
[ 4k4^2 - 16(4 - k)(2k1) 0 ]
This can be rewritten as:
[ 4(k4)^2 - 4(32k - 16 2k^2 - k) 0 ]
Further simplification leads to:
[ 36k^2 - 108k 0 ]
Step 3: Solving the Equation
We can factorize the above equation as follows:
[ 36k(k - 3) 0 ]
This gives us two solutions:
[ k 0 quad text{or} quad k 3 ]
Thus, the values of k that make the given quadratic equation a perfect square are (k 0) and (k 3).
Step 4: Verifying the Solutions
Let's verify these solutions by substituting them back into the original equation.
For (k 0):The equation becomes (4x^2 0 - 8 0), which simplifies to (4x^2 8), or (x^2 2), which is not a perfect (k 3):The equation becomes (4 - 3x^2 24x 24 0), or (-3x^2 24x 24 0). This can be rearranged as ((x - 5)^2 0), which is a perfect square.Therefore, the value of k that makes the given quadratic equation a perfect square is (k 3).
Conclusion
To summarize, in order for the quadratic equation (4 - kx^2 2k4x 8k1 0) to be a perfect square, the value of k must be 3. This is a key concept in the study of quadratic equations and is crucial for understanding more complex mathematical problems.