Question

How do you solve the $2{x^2} - 13x - 7 = 0?$

Answer 1

Hint: According to given in the question we have to determine the value of the variable x or we can say that we have to solve the given expression which is $2{x^2} - 13x - 7 = 0$. So, to determine the value of x for the given quadratic expression $2{x^2} - 13x - 7 = 0$ first of all we have to compare the given expression with the general form of the expression which is as mentioned below:
$ \Rightarrow $ General form of quadratic expression: $a{x^2} + bx + c = 0$……………….(A)
Now, we have to compare the given quadratic expression with the general form of the expression as mentioned above so that we can determine the values of a, b, and c.
Now, to find the roots or zeros of the expression we have to use the formula to find the roots or the zeroes of the quadratic expression which is as mentioned below:

Formula used:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}..............(B)$
Where, a is the coefficient of ${x^2}$, b is the coefficient of x and c is the constant term which is given in the quadratic expression.
Now, to determine the roots we have to substitute the values of a, b, and c which we have already obtained by comparing the given expression with the general form of the expression.

Complete step-by-step answer:
Step 1: First of all we have to compare the given expression with the general form of the expression (A) which is as mentioned in the solution hint. Hence, on comparing,
$
   \Rightarrow a = 2, \\
   \Rightarrow b = - 13, \\
 $
And,
$ \Rightarrow c = - 7$
Step 2: Now, to find the roots or zeros of the expression we have to use the formula (B) to find the roots or the zeroes of the quadratic expression which is as mentioned in the solution hint. Hence,
$ \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {{{( - 13)}^2} - 4 \times 2 \times ( - 7)} }}{{2 \times 2}}$…………………(1)
Step 3: Now, to obtain the value of x we just have to open the square and add and subtract the terms which can be in the expression (1) as we have obtained in the solution step 2. Hence,
$
   \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {169 + 56} }}{4} \\
   \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {225} }}{4} \\
 $
Now, we have to obtain the square root of 225 which is as obtained in the expression just above,
$ \Rightarrow x = \dfrac{{ - 2 \pm 15}}{4}...............(2)$
Step 4: Now, to obtain both of the roots with the help of the expression (2) which is as obtained in the solution step 3 we have to add and subtract the terms of the expression. Hence,
On adding the terms in the right hand side of the expression,
$
   \Rightarrow x = \dfrac{{ - 2 + 15}}{4} \\
   \Rightarrow x = \dfrac{{13}}{4} \\
 $
On subtracting the terms in the right hand side of the expression,
$
   \Rightarrow x = \dfrac{{ - 2 - 15}}{4} \\
   \Rightarrow x = \dfrac{{ - 17}}{4} \\
 $

Hence, with the help of the general form of the expression (A) and the formula (B) we have determined the solution for the given quadratic expression which is $x = \dfrac{{13}}{4}$ and $x = \dfrac{{ - 17}}{4}$.

Note:
To determine the roots of the quadratic expression it is necessary that we have to determine the values of a, b, and c which can be obtained by comparing the given expression with the general form of the expression.
On solving a quadratic expression only two roots or zeros can be obtained which will satisfy the given quadratic expression.