Solving – 3) = 0(?4 + 4)(?4 –

Solving quadratic equations might seem like a tedious task and the squares might come as a nightmare to the first timers but trust me it is not. Once you know the pattern, use the formula and mainly you practice, it is much fun to do that. As discussed in the last article, there are three ways of solving quadratic equations:1. Factorization2. Completing the square method3. Quadratic formula4.Let’s see each one in detail:Factorization:The name itself suggests, factorization, use of factors to get the value of x. Here, we find what to multiply to make the quadratic equation.Let’s see an example and we will get to know more about it,Example:Let’s consider the equation: x2 + 3x – 4 = 0Here, a=1, b=3, c=-4 so when we multiply (a x c) we get, 1 x -4 =-4,and this should give us b which is 3.In order to get 3, we can have (+4) and (-1) as the factors, whose multiplication is -4 and addition is 3.It can be written as: (x+4)(x-1)(x+4)=0 (x-1) =0 Set each factor to 0x+4=0 x-1=0 Solve these equationsx=-4 or x=1You can check these by placing the values in x in (x+4) (x-1); we get,Checking x = ?4 Checking x = 1(x + 4)(x – 3) = 0 (x + 4)(x – 3) = 0(?4 + 4)(?4 – 3) = 0 (3 + 4)(1-1) = 0(0)( ?7) = 0 (7)(0) = 00 = 0 0 = 0Hence the values are correct.One more way is, when the equation isExample: 2×2 + 7x + 3Here, a is not 1, the factors required, will be, the multiplication is (2 x 3) = 6 and the sum should be 7. The factors can be 1 and 6 and written as:2×2 + (1x + 6x) + 3 = 0 substituting the factors2x(x+1)+3(x+1)=0 Bringing the common out(2x+1)(x+1)=0(2x+1)=0 or (x+1)=0X=-1/2 or x=-1Well, it is not mandatory we get the results always by using this method. In that cases, we can use the below methods.Completing the square method:Let’s understand this with an example:Example:2×2=12x+54, we need to convert this to the standard form, we will get,2×2-12x-54=02×2/2-12x/2=54/2 Get all x terms on the left and divide it by a, i.e.2 to get a=12×2-6x+9=27+9 Complete the square and to do that add (b/2)2 to both sides?(x-3)2 = ?36 Write as binomial squaredx-3=±6 Take square root of both sidesx=(3+6) or (3-6)x=9 or x=-3Quadratic Formula:If we do not get the desired output from the above two, we will definitely get by this method. Herewe use the special Quadratic Formula:Just plug in the values of a, b and c, and do the calculations. We will look at this method in more detail now. The below image illustrates the best use of quadratic equation.Example:Here a=3,b=10 and c=-25By substituting the values in the formula we get the result.There is also a different scenario in this,When the Discriminant (the value b2 ? 4ac) is negative we get complex solutions. What does that mean? This is when imaginary numbers are involved.Example:5×2 + 2x + 1 = 0here a=5, b=2, c=1b2 ? 4ac = 22 ? 4×5×1 Discriminant is negative:-16x = ?2 ± ?(?16)10 Use the Quadratic Formula?(?16) = 4i (where i is the imaginary number ??1)x = ?2 ± 4i/10x = ?0.2 ± 0.4iThese methods will help you solve the quadratic equations.