Question

How do you solve ${{x}^{2}}-5x=14$ ?

Answer 1

In this question, we have to find the value of x. The equation given to us is in the form of a quadratic, therefore when we solve this problem, we will get two values for x, which satisfy the equation. Therefore, we will apply the discriminant method to solve this problem. We first find subtract 14 on both sides of the equation and then get the value of a, b, and c. Then, we will get the value of discriminant $D=\sqrt{{{b}^{2}}-4ac}$, and thus find the value of x using the discriminant formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .After necessary calculations, get two equations of x, we solve them separately to get the value of x, which is our required answer.

Complete step by step answer:
According to the question, a quadratic equation is given to us and we have to solve the equation for the value of x.
The equation is ${{x}^{2}}-5x=14$ ----------------- (1)
Firstly, we will subtract 14 on both sides of the equation (1), we get
$\Rightarrow {{x}^{2}}-5x-14=14-14$
As we know, the same term with opposite signs cancel out each other, therefore we get
$\Rightarrow {{x}^{2}}-5x-14=0$ ------------- (2)
As we know, the general quadratic equation is in form of $a{{x}^{2}}+bx+c=0$ ---------- (3)
Thus, on comparing equation (2) and (3), we get $a=1,$ $b=-5,$ and $c=-14$ ------- (4)
So, now we will find the discriminant $D=\sqrt{{{b}^{2}}-4ac}$ by putting the above values in the formula, we get
$\begin{aligned} & \Rightarrow D=\sqrt{{{(-5)}^{2}}-4.(1).(-14)} \\\ & \Rightarrow D=\sqrt{25-(-56)} \\\ \end{aligned}$
Thus, on further solving, we get
$\begin{aligned} & \Rightarrow D=\sqrt{25+56} \\\ & \Rightarrow D=\sqrt{81} \\\ \end{aligned}$
We know 9 is the square root of 81, we get
$\Rightarrow D=9$ -------------- (5)
Since we see the discriminant is a real number, thus now we will find the value of x, using the discriminant method,
$\Rightarrow x=\dfrac{-b\pm D}{2a}$
$\Rightarrow x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ --------------- (6)
So, we will put the value of equation (4) and (5) in equation (6), we get
$\Rightarrow x=\dfrac{-(-5)\pm 9}{2.(1)}$
On further simplification, we get
$\Rightarrow x=\dfrac{5\pm 9}{2}$
Therefore, we will split the above equation in terms of (+) and (-), we get
$\Rightarrow x=\dfrac{5+9}{2}$ -------- (7) , or
$\Rightarrow x=\dfrac{5-9}{2}$ ---------- (8)
Now, we will first solve equation (7), we get
$\Rightarrow x=\dfrac{14}{2}$
As we know, 2 is the factor of 14 and 2, therefore, we get
$\Rightarrow x=7$ -------- (9)
Now we will solve equation (8), we get
$\Rightarrow x=\dfrac{-4}{2}$
As we know, 2 is the factors of 2 and 4, therefore we get
$\Rightarrow x=-2$ -------- (10)
Thus, from equations (9) and (10), we get the value of x.
Therefore, for the equation ${{x}^{2}}-5x=14$ , we get the value of $x=7,-2$ .

Note:
While solving this problem, do all the steps carefully and avoid errors to get the correct answer. One of the alternative methods for solving this problem is using splitting the middle term method. In this method, we first split the middle term in the factors of the constant value and then take the common value among all and make necessary calculations to get the required value of x.