3 Easy Methods for Solving Quadratic Equations

Quadratic formula and the equations are an essential part of algebra. The equations can be solved by using four different methods. Yes there are quadratic equation calculators are available, but we don’t have access to such calculators all the time.  So, let’s discuss these methods one by one with an example.

Factoring method

The method requires a lot of practice as in it students need time to understand this method properly. In it, we solve expression 3x² - 11x - 4 = 0

Step 1: In the first step, we need to multiply 3 and 4 and then find out the LCM of its result. After multiplication, we get 12 and LCM 2, 2, 3 and 1. Now, there is a trick to solve the equation as you need to multiple these obtained numbers in such a way that it results in 11 when subtracted and 12 when multiplied.

3x² - 11x - 4 = 0

3x² - 12x + x - 4 = 0

In the above step, we have 12x and x in the middle of the equation. When we solve (-12x) and (-x), then we get (-11x). Just like that, we will get 12x² after multiplying 3x² and (4).

Step 2:  In the second step, we take 3x common and from the second half part 1 as a common number. Here what it looks like?

3x(x - 4) + 1(x - 4) = 0

(3x + 1) (x - 4) = 0

Step 3: In the last step, we need we will equate (3x + 1) and (x - 4) to 0. The two answers for variable X are (- 1⁄3) and 4.

Completing the square

The method is quite tricky however easy at the same time. We will solve equation

2x² - x – 1 = 0

By this method:

Step 1: At first we will divide the equation by 2 from both the sides and then the equation looks like this:

2x² - x – 1 = 0

Step 2: In next step, we need to add 1/2   from both the sides and this equation will be like this:  

x² - x⁄2 - 1⁄2 = 0

Step 3: In the above step, two numbers get canceled, i.e.,  after which the equation will reduce to:

x² - 1x⁄2 = 1⁄2

Step 4: In the fourth step, we will multiply the two numbers of the equation, i.e.

(1⁄2 * 1⁄2)² which results in in1/16.

x² - 1x⁄2 + 1⁄16 = 1⁄2 + 1⁄16

Step 5: Now, we simplify the RHS as it is an expansion of formula while you have to factorize the LHS.

(x - 1⁄4)² = 9⁄16

Step 6: Now in the last step, you need to square root both the sides to get the value of x. After that we will get:

x = 1⁄2 and 1.

Quadratic Formula

The easiest among all methods as just put the equation values in the formula to get answers. Here is the quadratic formula:

x = -b ± √b²-4ac⁄2a

We will solve expression 2x² - x -1 = 0, after putting values in the formula we will solve it in the following way:

x = - (-1) ± √ (-1)²-4(2) (-1) ⁄2 (-1)

Here the value of a = 2, b is (-1), and c is also (-1), make sure to put right sings along with the number. Sings play an important role as one single wrong sign will lead to wrong answers.

x = 1 ± √9⁄4

After which we will get two values of:

x = 1 ± 3⁄4

And we will get 2 values, i.e., x = 1 and x = (-1⁄2)

Along with this, you can even use quadratic formula calculator available on the internet. It is just like a calculator where you need to put numbers and get direct answers.

Learn and Use Quadratic Formula with an Example

Mathematics is quite tricky but an interesting subject.  It is full of equations and calculations that have much significant importance in different fields. Learning this subject is very useful to excel in various fields and also to clear many competitive exams.  So, today we will learn about a simple yet useful topic that is quadratic equations. To solve any quadratic equation you need to know its formula and steps to use the quadratic formula. In this article, you will get the introduction of quadratic equations along with its different forms and examples.

Quadratic Equation

Quadratic equation – its introduction

It is a second-degree equation that means it has at least one term with a square. The equation is always defined in its standard form. The standard form of this equation is ax² + box +c = 0 where a, b and c are the constants or coefficients. In this equation, there is a variable that is ‘X’ with an unknown value. Rest of the coefficients has known value. Using the quadratic formula, we usually find the value of this unknown variable. An equation is called quadratic if:

·         Its first variable that is ‘a’ is not equal to zero while the other variables can be zero.

·         The third variable is constant or an absolute term.

·         It must be a second-degree equation.

·         Value of unknown variable, i.e., x must satisfy the equation and hence called the root of an equation.

To understand what the quadratic equation is, here are examples of quadratic equation in its different form.

·         Standard form equation: 3x2+4x-12=0

·         Equation without the linear coefficient: x²-8=0

·         Equation without constant: 2x2+ 16x=0

·         Equation in factor form: (x+2) (x-4) =0

·         In another form: x(x-2) = 8, on multiplying and moving eight the equation comes in standard form and becomes x²-2x-8=0

These were some of the examples of quadratic equation in its different forms. Let’s see the procedure to solve these equations. Using a quadratic formula you can solve any quadratic equation, you can also use the other method where equate the factor to zero and get the value. But this method does not work for every problem as sometimes it can be quite messy because it does not get factored. The formula for the standard equation is:

                                                                     x= (-b±√ (b²-4ac))/2a

To use this formula, first, arrange the equation in the standard form.  Remember that the coefficient b will get squared completely means along with the sign. On solving the problem, you will get two roots, one with a positive value and other with the negative value. Take an example to understand the formula.

For example:

To solve x²+2x-4=0 , You can find the factor or simply put the values of a, b and c in the above formula to get the answer. Let’s solve using the quadratic formula.  The value of coefficients a, band c is 1, 2 and-4 respectively. Put them in the formula.

x= (-2±√ (2²-4.1.-4))/2.1

= (-2±√(4 +16))/2

= (-2±√20)/2

= (-2±2√5)/2   

The two roots of equation are -2+2√5/2 and -2-2√5/2

If you understand this example, then try solving other problem as well. Go with the simple problems and then move toward the tough questions. I hope you understand how to use the quadratic formula.