For many students, solving a quadratic equation is a challenge. While the process may seem intimidating, it can be mastered with practice. This article will provide a straightforward guide to solving a quadratic equation, with step-by-step instructions and helpful tips.

## What is a Quadratic Equation?

A quadratic equation is an equation of degree two with two unknowns or variables. It is usually written in the form ax^{2} + bx + c = 0, where a, b and c are constants and x is the unknown. It can be further divided into three categories:

- Standard Form: ax
^{2}+ bx + c = 0 - Factored Form: ax
^{2}+ bx + c = (x-m)(x-n) - Vertex Form: ax
^{2}+ bx + c = a(x-h)^{2}+k

## Step 1: Identify the Parts of the Equation

The first step in solving a quadratic equation is to identify the parts of the equation. Make sure to identify both the variables and constants. In standard form, the coefficients are usually written as a, b and c, while the unknown is written as x. In factored form, the constants are known as m and n, and in vertex form, they are known as h and k.

## Step 2: Rewrite the Equation in Standard Form

The next step is to rewrite the equation in standard form, if the equation is not presented in that form already. This can be done by combining like terms and rearranging the equation. In vertex form, this is done by adding a and c to both sides, while in factored form, this is done by multiplying out the brackets.

## Step 3: Use the Quadratic Formula to Solve for x

Once the equation has been written in standard form, it can be solved using the quadratic formula. This formula can be used to solve equations of degree two, and gives the two possible solutions to the equation. The quadratic formula is expressed as x = (-b ± √(b^{2} – 4ac))/2a, where a, b and c are the same constants from step 1.

## Step 4: Check Your Answer

Once you have calculated the solutions for x, check them to make sure they are correct. To do this, substitute each solution into the equation and check if it satisfies it. If both solutions do indeed solve the equation, then you have found your answers.

## Step 5: Graphical Representation of the Solution

To gain a better understanding of the solution, it can be helpful to draw a graph. This process involves plotting the x-intercepts (the solutions for x) on an x-y graph. The graph should intersect the x-axis at those points and should be shaped like a parabola, with its vertex at the maximum or minimum of the parabola.

## Step 6: Using Completing the Square to Solve a Quadratic Equation

Sometimes it can be helpful to rephrase a quadratic equation into a more simple form before using the quadratic formula. This process is known as completing the square and involves rewriting it in vertex form or factored form. Completing the square involves adding extra terms in order to turn it into a perfect square trinomial of the form ax^{2} + bx + c = a(x+h)^{2}+k.

## Step 7: Using Factoring to Solve a Quadratic Equation

A final way to solve a quadratic equation is to factor it into two separate equations and then solve for each one individually. Factoring a quadratic involves applying techniques such as grouping and differences of squares. If you are able to factor the equation into two separate equations, you can then solve each one individually.

## Tips for Solving Quadratic Equations

- Always start by rewriting the equation in standard form.
- Make sure you substitute your answer back into the original equation to check that it’s correct.
- Place special emphasis on learning how to complete the square.
- If possible, graph the solution on an x-y graph.
- Practice with simpler equations before attempting more complex ones.

## Common Mistakes to Avoid when Solving Quadratic Equations

- Including constants that are not part of the equation: Many students make the mistake of including constants that are not part of the equation when calculating their solutions.
- Forgetting to take account of signs: Students often forget to take account of plus and minus signs when calculating solutions for x.
- Solving for b instead of x: When trying to solve an equation quickly, some students forget that they have to solve for x and instead solve for b.
- Thinking that there are always two solutions: Some students assume that all quadratic equations have two solutions when this is not necessarily true.
- Neglecting to factor in special cases: Special cases such as perfect squares should always be factored in.

With practice and patience, any student can learn how to solve a quadratic equation. By following our seven steps and avoiding common mistakes, you will be able to find solutions with confidence.