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.
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.,
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.
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.