Question

Solve the following equations by the method of completion of a square. $5{x^2} - 24x - 5 = 0$

Answer 1

Hint: First, we need to know about the concept of the quadratic equations, which is in the form of at most degree two variables, like the general quadratic equation is $a{x^2} + bx + c = 0$
Since we need to convert the given equation in some form in generalized form then it is easy to solve the given in the completion of the square method. Also note that in the quadratic method there are several methods to solve the given problem like using the quadratic formula, factorization method, and completion of the square method.

Complete step by step answer:
Since given that the equation is of the form $5{x^2} - 24x - 5 = 0$
Using the division operation, divide all the terms by the number $5$ then we get $\dfrac{{5{x^2}}}{5} - \dfrac{{24x}}{5} - \dfrac{5}{5} = 0$ and thus we have $\dfrac{{5{x^2}}}{5} - \dfrac{{24x}}{5} - \dfrac{5}{5} = 0 \Rightarrow {x^2} - \dfrac{{24x}}{5} - 1 = 0$
Now Turing the constant value into the right side, note that while changing the position of the values, the sign of the values or numbers will change and hence we get ${x^2} - \dfrac{{24x}}{5} = 1$
Now equally adding the number ${(\dfrac{{24}}{{10}})^2}$ on both sides then we have ${x^2} - \dfrac{{24x}}{5} + {(\dfrac{{24}}{{10}})^2} = 1 + {(\dfrac{{24}}{{10}})^2}$
Using the formula that ${a^2} - 2ab + {b^2} = {(a - b)^2}$ then we have ${x^2} - \dfrac{{24x}}{5} + {(\dfrac{{24}}{{10}})^2} = 1 + {(\dfrac{{24}}{{10}})^2} \Rightarrow {(x - \dfrac{{24}}{{10}})^2} = 1 + {(\dfrac{{24}}{{10}})^2}$
Since ${(\dfrac{{24}}{{10}})^2} = 5.76$ using division and square and then we get $ \Rightarrow {(x - \dfrac{{24}}{{10}})^2} = 1 + 5.76$
By the addition we get ${(x - \dfrac{{24}}{{10}})^2} = 6.76$
Taking square root on both sides then we get ${(x - \dfrac{{24}}{{10}})^2} = 6.76 \Rightarrow {\sqrt {(x - \dfrac{{24}}{{10}})} ^2} = \sqrt {6.76} \Rightarrow (x - \dfrac{{24}}{{10}}) = \sqrt {6.76} $
Hence further solving we get $(x - \dfrac{{24}}{{10}}) = \sqrt {6.76} \Rightarrow x - 2.4 = \pm 2.6$
Thus, we have $x = \pm 2.6 + 2.4 \Rightarrow x = 5,x = - 0.2$ hence which is the required value.

Note:
Also, note that in the quadratic $a{x^2} + bx + c = 0$ the $a$ will never be zero if $a = 0$ then $bx + c = 0$ turns into a linear one-degree equation.
Taking the square root to the positive values get the outcome as positive or negative
We also make use of the basic operations, The addition is the sum of given two or more than two numbers, or variables and in addition, if we sum the two or more numbers then we obtain a new frame of the number will be found, also in subtraction which is the minus of given two or more than two numbers, but here comes with the condition that in subtraction the greater number sign represented in the number will stay constant example $2 - 3 = - 1$
The other two operations are multiplication and division operations.