How to Use the Quadratic Equation in Easy Method?

The quadratic equation is a fundamental concept in mathematics that used to solve equations that have the form ax^2 + ax + c = 0, where a, b, and c are coefficients. The quadratic equation has a number of applications in science, engineering, and finance, and is therefore an important tool for students to learn.

Methods for solving quadratic equations

One of the most straightforward methods for solving quadratic equations is to use the quadratic formula, which given by:

x = (-b ± sqrt (b^2 - 4ac)) / 2a

However, this formula can sometimes be difficult to use, especially if the values of a, b, and c are large or complicated. A more user-friendly method for solving quadratic equations is the method of completing the square.

Completing the square is a technique for transforming a quadratic equation of the form ax^2 + bx + c = 0 into a perfect square trinomial. To use this method, follow these steps:

Step 1

Divide both sides of the equation by a Dividing both sides of the equation by a simplifies the equation to the form x^2 + (b/a) x + (c/a) = 0.

Step 2

Move the constant term to the other side of the equation to complete the square, we need to isolate the x terms on one side of the equation. Therefore, we can move the constant term (c/a) to the other side of the equation by subtracting it from both sides.

x^2 + (b/a) x = - (c/a)

Step 3 

Add the square of half the coefficient of x to both sides of the equation to complete the square, we need to add a term to both sides of the equation that will make the left-hand side a perfect square trinomial. This term is the square of half the coefficient of x, which is (b/2a) ^2. We add this term to both sides of the equation:

x^2 + (b/a) x + (b/2a) ^2 = (b/2a) ^2 - (c/a)

The left-hand side of the equation is now a perfect square trinomial, which can factor as (x + b/2a) ^2. Simplifying the right-hand side of the equation gives:

(x + b/2a) ^2 = (b^2 - 4ac) / 4a^2

Step 4

Take the square root of both sides of the equation Taking the square root of both sides of the equation gives:

x + b/2a = ± sqrt ((b^2 - 4ac) / 4a^2)

Step 5

Solve for x Finally, we can solve for x by subtracting b/2a from both sides of the equation and simplifying:

x = (-b ± sqrt (b^2 - 4ac)) / 2a

As we can see, Completing the square method helps us simplify the quadratic equation into a perfect square trinomial and therefore solving the equation by finding the roots. This method can be particularly useful when the quadratic equation is not in the standard form of ax^2 + bx + c = 0, or when we interested in finding the vertex of the parabola.

Disadvantage

One disadvantage of the Completing the square method is that it can be more time-consuming than the quadratic formula, especially for larger values of a, b, and c. However, completing the square can be a useful method to know for solving quadratic equations, especially when dealing with non-standard equations.

Important application

Another important application of completing the square is finding the vertex of the parabola represented by the quadratic equation. Recall that the vertex is the point on the curve where it changes direction. If the quadratic equation is in standard form, then the vertex is easy to find since it is simply (-b/2a, f(-b/2a)).

However, if the equation is not in standard form, we can use  Completing the square to rewrite it in vertex form, which is y = a (x - h) ^2 + k, where (h, k) is the vertex. For example, consider the quadratic equation 3x^2 + 12x + 10 = 0.

Dividing by 3 to get the coefficient of x^2 equal to 1, we get x^2 + 4x + 10/3 = 0. Moving the constant term to the other side of the equation and completing the square gives us (x + 2) ^2 + 2/3 = 0.

Now, we can see that the vertex is (-2, 2/3). To verify this, we can rewrite the equation in vertex form as y = 3(x + 2) ^2 + 2. This form makes it clear that the vertex is (-2, 2/3), and the coefficient of x^2 tells us whether the parabola opens up or down.

Conclusion

In conclusion, the quadratic equation is an essential tool in mathematics and has many practical applications. Although the quadratic formula is a powerful tool for solving quadratic equations, it can sometimes be difficult to use. The method of Completing the square is a more user-friendly method for solving quadratic equations and can be especially helpful in situations where the quadratic equation is not in standard form. By following the steps outlined above, students can learn to use the completing the square method to solve quadratic.