Quadratic equations are a fundamental part of algebra and have a wide range of applications in various fields, including physics, engineering, and economics. This article will delve into the factors of the quadratic equation (x^2 - 9x 21 0) and explore the methods to solve it.

Introduction to Quadratic Equations

A quadratic equation is an equation of the second degree, typically in the form of (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). The maximum degree of a quadratic equation is 2, and it can be solved using various methods such as factoring, completing the square, or using the quadratic formula.

Factoring Quadratic Equations

Factoring a quadratic equation involves rewriting the equation as a product of two binomials. This method is particularly useful when the quadratic equation has rational roots. However, not all quadratic equations are easily factorable.

The Given Equation: (x^2 - 9x 21 0)

Let's consider the equation (x^2 - 9x 21 0). The coefficient of (x^2) is 1, which simplifies the factoring process. To factor this equation, we need to find two numbers that multiply to 21 and add to (-9).

No Rational Roots Exist

After a thorough inspection, it is evident that there are no two numbers that multiply to 21 and add to (-9). This means that the given quadratic equation does not have rational roots. Therefore, the equation cannot be easily factored into simpler linear factors.

Alternative Methods to Solve the Equation

Since factoring is not straightforward, we can use alternative methods such as completing the square or the quadratic formula to solve the equation.

Using the Quadratic Formula

The quadratic formula is given by:

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

For the equation (x^2 - 9x 21 0), we have (a 1), (b -9), and (c 21).

Substituting these values into the quadratic formula:

$$x frac{-(-9) pm sqrt{(-9)^2 - 4(1)(21)}}{2(1)}$$

$$x frac{9 pm sqrt{81 - 84}}{2}$$

$$x frac{9 pm sqrt{-3}}{2}$$

Since the discriminant (Delta -3), the roots are complex numbers. The square root of a negative number results in an imaginary number. We can express (sqrt{-3}) as (sqrt{3}i).

Therefore, the roots are:

$$x frac{9 pm sqrt{3}i}{2}$$

So, the solutions to the quadratic equation are:

$$x frac{9 sqrt{3}i}{2} quad text{or} quad x frac{9 - sqrt{3}i}{2}$$

Conclusion

Quadratic equations can be solved using various methods, including factoring (when possible), completing the square, or using the quadratic formula. While the given equation (x^2 - 9x 21 0) cannot be easily factored, we can use the quadratic formula to find its roots, which are complex numbers.

Understanding these methods is crucial for solving a wide range of algebraic problems and is a fundamental skill in mathematics.