Question

How do you solve $ {{x}^{2}}+7x+10=0 $ ?

Answer 1

Hint: The given equation is a quadratic equation of the form $ a{{x}^{2}}+bx+c=0 $ . We will solve the equation by splitting the middle term such that the product of two terms is equal to $ a\times c $ and their sum or difference is equal to $ b $ .

Complete step by step answer:
We have been given an equation $ {{x}^{2}}+7x+10=0 $
We have to solve the given equation and find the value of the unknown variable.
We know that the general form of a quadratic equation is $ a{{x}^{2}}+bx+c=0 $ . The given equation is of the standard form so we will use the middle term split method to solve the given equation.
Now, we can split the middle term of the equation $ {{x}^{2}}+7x+10=0 $ as $ 5x+2x $ such that their sum is equal to $ 7x $ and product is equal to $ 10 $ .
Now, solving further we will get
 $ \begin{align}
  & \Rightarrow {{x}^{2}}+7x+10=0 \\
 & \Rightarrow {{x}^{2}}+5x+2x+10=0 \\
\end{align} $
Now, taking the common terms out we will get
 $ \Rightarrow x\left( x+5 \right)+2\left( x+5 \right)=0 $
Now, again taking the common terms out we will get
 $ \Rightarrow \left( x+5 \right)\left( x+2 \right)=0 $
Now, equating each factor to 0 we will get
 $ \begin{align}
  & \Rightarrow x+5=0 \\
 & \Rightarrow x=-5 \\
\end{align} $
And
 $ \begin{align}
  & \Rightarrow x+2=0 \\
 & \Rightarrow x=-2 \\
\end{align} $
Hence on solving the equation $ {{x}^{2}}+7x+10=0 $ we get the values of x as $ -5 $ and $ -2 $ .

Note:
 In this question, we get the two factors easily to obtain the product equal to $ a\times c $ and sum or difference equal to $ b $. Else we can use different methods like the quadratic formula method, factorization method, completing the square method, to solve a quadratic equation. We can also verify the value of x by substituting the obtained value in the given equation.