Question

Solve the following equations,
1) $3{x^2} = - 11x - 10$
2) $25x\left( {x + 1} \right) = - 4$

Answer 1

Hint: In order to solve this question, we have to write what is given in this question or what is mentioned in this question. Then we have to use the middle term splitting method to find the roots of the question. This will give us a clear picture of what our approach should be.

Complete step-by-step answer:
It is given in the question we have to solve $3{x^2} = - 11x - 10$ and $25x\left( {x + 1} \right) = - 4$ equations.
In the 1) part we should transfer $ - 11x - 10$ on the other side to make things simple
$ \Rightarrow 3{x^2} + 11x + 10 = 0$
Now, using the middle term splitting method we get,
$ \Rightarrow 3{x^2} + 6x + 5x + 10 = 0$
Now, taking out the common terms we get,
$ \Rightarrow 3x\left( {x + 2} \right) + 5\left( {x + 2} \right) = 0$
$ \Rightarrow \left( {3x + 5} \right)\left( {x + 2} \right) = 0$
To get values for ‘x’ as this is a quadratic equation we put them both equal to 0
$
  3x + 5 = 0 \\
 \Rightarrow 3x = - 5 \\
  \Rightarrow x = \dfrac{{ - 5}}{3} \\
 $
Or
$
  x + 2 = 0 \\
  \Rightarrow x = - 2 \\
 $
Therefore $\dfrac{{ - 5}}{3},2$ are the roots of the equation.
In in the 2) we multiply 25x by (x+1) and transpose -4 on the other side
 $
   \Rightarrow 25{x^2} + 25x = - 4 \\
   \Rightarrow 25{x^2} + 25x + 4 = 0 \\
 $
Now, using middle term splitting we get,
$ \Rightarrow 25{x^2} + 5x + 20x + 4 = 0$
Now, taking out the common terms we get,
$
   \Rightarrow 5x\left( {5x + 1} \right) + 4\left( {5x + 1} \right) = 0 \\
   \Rightarrow \left( {5x + 4} \right)\left( {5x + 1} \right) = 0 \\
 $
 To get values for ‘x’ as this is a quadratic equation we put them both equal to 0

$
  5x + 4 = 0 \\
 \Rightarrow 5x = - 4 \\
  \Rightarrow x = \dfrac{{ - 4}}{5} \\
 $
Or
$
  5x + 1 = 0 \\
\Rightarrow 5x = - 1 \\
\Rightarrow x = \dfrac{{ - 1}}{5} \\
 $
  Therefore $\dfrac{{ - 4}}{5},\dfrac{{ - 1}}{5}$ are the roots of the equation,

Note: Whenever we face such a question the key concept is first that we should write what is given in this question like we did in this question. This question is solved by the middle term splitting method of quadratic equations. First, we have to transpose the terms on the other side in 1) part whereas in 2) part we have to multiply by the equation. Also we have to take the terms common to get the last part and then we have to put the equations in last equal to 0. This will help us to solve questions easily.