Solve Quadratic Equation Calculator

Quadratic equations are a fundamental concept in algebra and mathematics. They are used to model various real-world situations, such as projectile motion, electrical circuits, and optimization problems. In this article, we will discuss how to solve quadratic equations using the completing the square method.

To start with, let's consider the general form of a quadratic equation: x^2 + bx + c = 0, where x is the variable, b is the coefficient of x, and c is the constant term. To solve this equation, we can use the following steps:

1. If possible, try to factor the equation into two binomials. This will make it easy to solve for x.

2. If factoring is not possible, we can use the completing the square method.

The completing the square method involves adding a constant term to both sides of the equation such that the left-hand side becomes a perfect square trinomial. The steps are as follows:

1. Take half of the coefficient of x and square it.

2. Add this squared value to both sides of the equation.

3. Simplify the right-hand side by combining like terms.

Let's consider some examples to illustrate how to use the completing the square method.

Example 1: Solve x^2 + 6x + 5 = 0

To start with, we can rewrite the equation as (x + 3)^2 = 4. This is a perfect square trinomial, so we can take the square root of both sides to get x + 3 = ±√4 or x = -3 ± 1.

Example 2: Solve x^2 + 4x + 4 = 0

In this case, we can rewrite the equation as (x + 2)^2 = 0. This is also a perfect square trinomial, so we can take the square root of both sides to get x + 2 = 0 or x = -2.

In general, if b^2/4 < c, there are two solutions for x and you can find them by taking the square root of both sides. If b^2/4 = c, then we have one solution, which is equal to -b/2. If b^2/4 > c, then there's no solution.

To get from the right-hand side to the left-hand side, we need to perform the same operation on the right side. This involves subtracting c and adding b^2/4.

If the leading coefficient of your quadratic equation is not 1 (i.e., if the polynomial is not monic), then divide both sides by a to make it monic.

By following these steps, you can solve any quadratic equation using the completing the square method. This method is useful when factoring is not possible or when you need to find the exact solutions of the equation.

In conclusion, solving quadratic equations using the completing the square method requires some algebraic manipulation and attention to detail. However, with practice and patience, you can master this method and solve a wide range of quadratic equations.

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