Solving Quadratic Equations: Techniques and Applications

Solving quadratic equations is a fundamental skill in algebra that is widely used in various fields, from physics to engineering. There are several methods to solve quadratic equations, including direct factorization, completing the square, and using the quadratic formula. This article will explore each of these techniques and provide detailed examples of their application.

Direct Factorization

Direct factorization is the most efficient method when the quadratic equation can be factored directly. This method is particularly useful when the roots of the equation are rational and do not involve imaginary or radical numbers.

Example 1

Solve the equation: x^2 - 2x - 35 x5x - 7.

Solution:

ngather tx^2 - 2x - 35 0 t(x^2 - 7x 5x - 35 0) text{Group the terms:} (x(x - 7) 5(x - 7) 0) text{Factor out the common term:} ((x - 7)(x 5) 0) text{Set each factor to zero:} (x - 7 0) or (x 5 0) text{Solve for x:} (x 7) or (x -5)

The roots of the equation are (x 7) and (x -5).

Example 2

Solve the equation: x^2 - 9 x3x - 3.

Solution:

ngather tx^2 - 9 0 t((x - 3)(x 3) 0) text{Set each factor to zero:} (x - 3 0) or (x 3 0) text{Solve for x:} (x 3) or (x -3)

The roots of the equation are (x 3) and (x -3).

Direct Root Extraction

Another simple method is direct root extraction. This method is used when the quadratic equation can be simplified to the form x^2 c, where c is a constant.

Example 3

Solve the equation: x^2 7.

Solution:

ngather t(x^2 7) t(x pmsqrt{7})

The roots of the equation are (x sqrt{7}) and (x -sqrt{7}).

Completing the Square

Completing the square is a method that can be used to solve any quadratic equation, especially when direct factorization is not straightforward. This method is particularly useful when the coefficient of the x^2 term is 1.

Standard Form of a Quadratic

The standard form of a quadratic equation is ax^2 bx c 0. To complete the square, the equation is transformed into a perfect square trinomial.

Example 4

Solve the equation: x^2 8x 12 0.

Solution:

ngather tx^2 8x 12 0 text{Transform the equation:} (x^2 8x -12) (x^2 8x 16 -12 16) ((x 4)^2 4) text{Take the square root of both sides:} (x 4 pmsqrt{4}) text{Solve for x:} (x -4 pm 2) text{So, the roots are:} (x -2) and (x -6)

The roots of the equation are (x -2) and (x -6).

The Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation and is given by:

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

This formula is particularly useful when direct factorization or completing the square are not feasible.

Example 5

Solve the equation: 6x^2 - x - 12 0.

Solution:

ngather t6x^2 - x - 12 0 t(a 6, b -1, c -12) t(x frac{-(-1) pm sqrt{(-1)^2 - 4(6)(-12)}}{2(6)}) t(x frac{1 pm sqrt{1 288}}{12}) t(x frac{1 pm sqrt{289}}{12}) t(x frac{1 pm 17}{12}) text{So, the roots are:} (x frac{18}{12} frac{3}{2}) and (x frac{-16}{12} -frac{4}{3})

The roots of the equation are (x frac{3}{2}) and (x -frac{4}{3}).

Factoring Practice

Factoring quadratic equations into the form (ax - p)(bx - q) 0 also helps in identifying the roots directly. This method is particularly useful when the coefficients are integers.

Example 6

Solve the equation: 6x^2 - x - 12 0.

Solution:

ngather t6x^2 - x - 12 0 t((2x - 3)(3x 4) 0) text{The roots are:} (2x - 3 0) or (3x 4 0) text{So, the roots are:} (x frac{3}{2}) and (x -frac{4}{3})

Conclusion

There are multiple methods to solve quadratic equations, each with its own advantages. Direct factorization, completing the square, and the quadratic formula are all valuable tools in a problem-solver's arsenal. Understanding these techniques will help you tackle a wide range of problems with ease.