Question

Solve the following quadratic equation :-${{x}^{2}}+7x-10=0$.

Answer 1

Hint: The equation given in the question is a quadratic equation.The general form of the quadratic equation is given as $a{{x}^{2}}+bx+c=0$. So, we will be using the middle-term split method i.e.finding 2 factors such that the product is $a\times c$ and their sum or difference is $b$ and find the value of $x$.

Complete step-by-step answer:

The equation given in the question is $-{{x}^{2}}+7x-10=0$.
We can solve this by using the middle-term split method. In this method, we will factorise the middle term, that is the term containing $x$.
The general form of a quadratic equation is $a{{x}^{2}}+bx+c=0$. The middle term is $bx$. To get the factors of $bx$, we have to first find the sum and product of the factors. So, the product of the factors is given by $a\times c$. After that we have to find 2 factors such that their product is $a\times c$ and their sum or difference is $b$.
Now, let us consider the equation in the given question, $-{{x}^{2}}+7x-10=0$,
Hence, comparing with the general form, we get $a=-1,b=7,c=-10$.
The product of the factors can be obtained as $a\times c\Rightarrow \left( -1 \right)\times \left( -10 \right)=10$
The sum of the factors should be $b$, which is $7$.
So, now we have to find two factors such that their product is $10$ and their sum or difference is $7$.
Let us consider the factors of $10$. We know that $10$ has four factors, $1,10,2,5$. We can see that when the factors $2$ and $5$ are added, it gives $2+5=7$, which is $b$.
So, we get the two factors as $2$ and $5$, so we can write the equation as,
$-{{x}^{2}}+2x+5x-10=0$
Taking the common terms out, we get,
$-x\left( x-2 \right)+5\left( x-2 \right)=0$
Taking $\left( x-2 \right)$ out, we get,
$\begin{align}
  & \left( -x+5 \right)\left( x-2 \right)=0 \\
 & \Rightarrow \left( 5-x \right)\left( x-2 \right)=0 \\
\end{align}$
Equating each factor to $0$, we get,
$\begin{align}
  & 5-x=0\Rightarrow x=5 \\
 & x-2=0\Rightarrow x=2 \\
\end{align}$
Therefore, we have obtained the values of \[x\] as $5$ and $2$.

Note: We have been asked to solve the quadratic equation in the question to find the value of $x$. We have different methods like the middle-term split method, using quadratic formula, completing the square method. We can use any one of these methods to obtain the value of $x$. Once we get the value of $x$, we can cross-check by substituting it in the given equation and make sure that the value satisfies the equation.