Introduction to Real and Equal Roots in Quadratic Equations

A quadratic equation in the form ( ax^2 bx c 0 ) can have real and equal roots under specific conditions. This article explores how to find the value of ( k ) that makes the equation ( x^2 - kx 4 0 ) have real and equal roots. We will use the concept of the discriminant to achieve this.

The Concept of Discriminant

The discriminant, denoted as ( D ), for a quadratic equation ( ax^2 bx c 0 ) is given by ( D b^2 - 4ac ). The discriminant plays a crucial role in determining the nature of the roots of a quadratic equation.

For Real and Equal Roots, D 0

For a quadratic equation to have real and equal roots, the discriminant must be zero. This condition ensures that the quadratic equation has exactly one real root (a repeated root). Therefore, for the quadratic equation ( x^2 - kx 4 0 ), we need to set the discriminant equal to zero:

[ D 0 ]

Calculating the Discriminant for ( x^2 - kx 4 0 )

Using the coefficients from the given quadratic equation, we have:

( a 1 ) ( b -k ) ( c 4 )

Substituting these values into the discriminant formula:

[ D b^2 - 4ac ]

Yields:

[ D (-k)^2 - 4 cdot 1 cdot 4 ]

Setting the discriminant equal to zero:

[ (-k)^2 - 16 0 ]

Solving for ( k ):

[ k^2 - 16 0 ]

[ k^2 16 ]

Therefore, the values of ( k ) are:

[ k 4 , text{or} , k -4 ]

Alternative Approach: Using Sum and Product of Roots

Another approach to solving this problem involves using the sum and product of the roots of the quadratic equation. According to Vieta's formulas, for the quadratic equation ( ax^2 bx c 0 ):

The sum of the roots ( alpha beta -frac{b}{a} ) The product of the roots ( alpha beta frac{c}{a} )

Given the quadratic equation ( x^2 - kx 4 0 ), let's denote the roots as ( alpha ) and ( beta ).

The product of the roots is:

[ alpha beta frac{c}{a} frac{4}{1} 4 ]

Since the roots are real and equal, let's consider ( alpha beta R ).

[ R^2 4 ]

Thus, ( R ) can be either 2 or -2.

Using the sum of the roots:

[ alpha beta R R 2R ]

From the equation ( x^2 - kx 4 0 ), the sum of the roots is:

[ alpha beta frac{-b}{a} k ]

Setting the sum of the roots to ( k ):

[ 2R k ]

For ( R 2 ):

[ 2 cdot 2 k Rightarrow k 4 ]

For ( R -2 ):

[ 2 cdot (-2) k Rightarrow k -4 ]

Therefore, the values of ( k ) are:

[ k 4 , text{or} , k -4 ]

Conclusion

By applying the concepts of the discriminant and Vieta's formulas, we have determined that the values of ( k ) for which the quadratic equation ( x^2 - kx 4 0 ) has real and equal roots are ( k 4 ) and ( k -4 ).

Understanding these concepts can be very useful in solving similar problems and in further studies of algebraic equations.