Ways of Solving Quadratic Equations

A quadratic equation is a polynomial of the highest degree two. That means that we will not find any term in the equation where ‘x’ is squared. By convention, the general form of a quadratic equation is ax² + bx + c = 0. a, b, and c are all numbers here, but with one exception, i.e., a cannot be zero. For example, the equation 3x² + 2x + 1 = 0 is in such a form, with ‘a being 3, ‘b’ being 2, and let a= 3, b = 2, and c = 1.

In the solution of a quadratic equation, we are concerned with finding x that will make the equation true. Such values are called the solutions of the equation. 1

Method 1: Use AI Homework Helper

If you don’t want to go by the conventional means, you can do it with the AI Homework Helper!

For students who want to double their study time, AI Homework Helper provides help on a variety of school topics by providing instant, step-by-step solutions to the most difficult questions. Its interface is very user-friendly, and you can enter any component by simply using the drag-and-drop concept. To capture an image of your homework, it’s a way for you to learn the concepts at a time suitable for you, and with the help that meets your expectations.

Sure. Here is a simple process of process you can use to solve the Quadratic Equation.

3x² + 2x + 1 = 0

Step 1: Prepare Your Problem

Take a photo of the “3x² + 2x + 1 = 0” equation. Take a photo of the “3x² + 2x + 1 = 0” equation that you’d like to solve from a textbook, your notebook, or your desktop computer screen and then transfer the photo to your Android phone.

Step 2: Upload the Image

On the AI Homework Helper site, click on the “Choose a File” button and upload the picture you just took.

Step 3: Ask Your Question

Then write/copy something simple like “Solve this equation” in the input box.

Step 4: Get the Solution

Click on the “Get Answer” button. In seconds, the AI will provide you with the complete answer.

Step 5: Understand the Explanation

Look at the step-by-step solution below that tells you everything you want to know about the process used in solving the equation, ranging from the quadratic formula to the answer. Take it a notch higher by even declaring your solution with more questions until you’re content.

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Method 2: Factoring to Solve

Factoring is a process whereby we take a quadratic expression and write it as a product of two expressions (which we call factors) that multiply together to yield the original quadratic expression. This method is based upon the use of the “Zero Product Property.”. This characteristic reads that as long as you’re working with two or more items that have been multiplied together, if their product is zero, then one of the items would need to be zero.

Let’s consider an example: A quadratic equation, say, x² + 5x + 6 = 0. To factorize this, we need to find two numbers which, when multiplied together, give 6 (the constant) and, when added together, give 5 (the coefficient of ‘x’). These are the numbers 2 and 3. Using these numbers, we can express the equation in factored form as (x + 2)(x + 3) = 0.

Now applying the Zero Product Property, we know that for this product to be zero, either (x + 2) = 0 or (x + 3) = 0..5 Setting each factor equal to zero, we have two simple linear equations: x + 2 = 0 or x + 3 = 0. Solving these equations, we have the solutions to the original quadratic equation as x = -2 or x = -3.5. Factoring is generally the quickest way of solving a quadratic equation, especially if the quadratic expression is not too hard to factor.5 However, it is important to note that not all quadratic equations can be easily factored with integer numbers.5

Method 3: Solving by Completing the Square

Another helpful way of solving quadratic equations is by using the method of completing the square. The trick here is to rearrange your quadratic equation in such a way that one side of the equation has a squared term by itself on one side and a constant on the other.

Let us demonstrate this with the help of an example: Solve the equation x² – 6x – 7 = 0. Begin by transferring the constant number (-7) to the other side of the equation by adding 7 to both sides. This gives us x² – 6x = 7. 31. Now, we would like to tackle the ‘x’ term. We take half of its coefficient (-6), getting -3, and then square it: (-3)² = 9. 6 This number, 9, is what we need to add to both sides of the equation in order to “complete the square” on the left side: x² – 6x + 9 = 7 + 9, which is x² – 6x + 9 = 16. 6 The left side of the equation, x² – 6x + 9, is now a perfect square trinomial, and we can factor it as (x – 3)². 5 So, our equation is now (x – 3)² = 16. To solve for ‘x’, we take the square root of both sides of the equation: x – 3 = ±√16, i.e., x – 3 = ±4. 5 Finally, we solve for ‘x’ by adding 3 to both sides: x = 3 ± 4. This gives us two possible solutions: x = 3 + 4 = 7 or x = 3 – 4 = -1. 5 Completing the square is a method that will work for all quadratic equations, even those that are difficult or impossible to factor easily. 6 Besides, the technique of completing the square is actually used to derive the very useful quadratic formula.

Method 4: Solving using the Quadratic Formula

If you have a quadratic equation in the form ax² + bx + c = 0, you can use the quadratic formula. It is a simple method of solving these types of equations by finding their roots. To have a clearer picture, let’s try an example: Let’s find the solution to the quadratic equation 2x² + 5x – 3 = 0. Then we can find the roots via the quadratic formula. Let’s take the numbers we have: a = 2, b = 5, and c = -3. We then substitute the values into the quadratic formula to find x. It will be x = negative 5 plus or minus the square root of 5 squared minus 4 times 2 times -3, divided by 2 times 2. Let us simplify the expression step by step now: we have x = [-5 ± √(25 + 24)] / 4, which then simplifies to x = [-5 ± √49] / 4. The square root of 49 being 7, our expression is now x = [-5 ± 7] / 4. We have two options to solve ‘x’. In case of addition: ‘x’ is the result of adding -5 and 7, and then dividing by 4, which equals one-half. Now, let us consider the example when we have a positive sign in the equation: x = (-5 + 7) / 4, which equals 2 / 4, which yields x = 0.

One tidy thing about the quadratic formula is the expression under the square root symbol, b² – 4ac. The discriminant is a helpful tool that gives us some information about the solutions without needing to solve the whole equation. When the discriminant (b² – 4ac) is more than zero, the equation has two different real solutions. If the discriminant is zero, then the equation possesses a single real solution referred to as a repeated root. Conversely, when the discriminant is negative, the equation has no real solutions but rather complex numbers as solutions. The quadratic formula is a reliable method that works everywhere and invariably provides the solution(s) to a quadratic equation if the solutions exist. 

Method 5: Graphing

To solve a quadratic equation graphically, you can plot the equation and determine where it crosses the x-axis. You likely know the routine: begin with the quadratic equation ax² + bx + c = 0. Then graphically plot it as y = ax² + bx + c. You can read off the x-values where it crosses the x-axis from the graph.

These precise points where the graph touches the x-axis are referred to as x-intercepts. The x-values of the x-axis are the solutions or roots of the equation ax² + bx + c = 0. The graph of a quadratic function is a parabola with its axis of symmetry parallel to the y-axis. In the case of a quadratic curve, the graph is a parabola in shape.

Conclusion

The quadratic formula is a ready reckoner to determine the roots of any quadratic equation given, once the equation is written in the form of a standard quadratic equation ax^2 + bx + c = 0. The formula is usually written as: x = [-b ± √(b² – 4ac)] / 2a.

Let’s see how to use it with the help of an example: Example Solve the quadratic equation 2x² +.

We’ve discussed four main methods of solving quadratic equations in this post: factoring, completing the square, the quadratic formula, and graphing. 5 Factoring can be a good method when the expression is a simple factorable quadratic. 5 Completing the square is a method that always works (and is required if you are to derive the quadratic formula), and gives you an insight into why the quadratic formula works. 6 The quadratic equation The quadratic formula is possibly the most widely trusted method, as it will work for quite literally any quadratic equation to determine its solutions. 6 Finally, but not least importantly, a graphical interpretation of the solutions as the x-intercepts of the related quadratic functions.

Comparison of Methods for Solving Quadratic Equations