Quadratic Equation with Given Roots: -1 and -3

The problem of finding a quadratic equation whose roots are -1 and -3 can be approached in several ways. Let's explore the steps and the reasoning behind the solution.

General Form of a Quadratic Equation

A quadratic equation in standard form is:

[ x^2 - (alpha beta)x alphabeta 0 ]

Given the roots (alpha -1) and (beta -3), we can substitute these values into the equation.

Deriving the Quadratic Equation

Let's follow the steps to derive the quadratic equation:

The sum of the roots, ((alpha beta)), is:

[ (-1) (-3) -4 ]

The product of the roots, (alphabeta), is:

[ (-1) times (-3) 3 ]

The quadratic equation with roots -1 and -3 can be written as:

[ x^2 - (text{Sum of roots})x (text{Product of roots}) 0 ]

Therefore, the quadratic equation becomes:

[ x^2 - (-4)x 3 0 ]

This simplifies to:

[ x^2 4x 3 0 ]

General Form of Quadratic Equation with Given Roots

It is important to note that any scalar multiple of the equation (x^2 4x 3 0) will also be a valid quadratic equation with the same roots. For example, if (a) is any non-zero constant, the equation:

[ a(x^2 4x 3) 0 ]

will still have the same roots -1 and -3.

Alternative Approach: Factoring

We can also verify the same result using the factored form of the quadratic equation:

The quadratic equation with roots -1 and -3 can be factored as:

[ (x - (-1))(x - (-3)) 0 ]

This simplifies to:

[ (x 1)(x 3) 0 ]

Expanding the factored form:

[ x^2 3x x 3 x^2 4x 3 0 ]

Conclusion

In conclusion, the quadratic equation whose roots are -1 and -3 is:

[ x^2 4x 3 0 ]

If a is any non-zero constant, the equation:

[ ax^2 4ax 3a 0 ]

will also have the roots -1 and -3.