Question

How to solve this? $ 2x + {x^2} \geqslant 35 $

Answer 1

Hint: As in the question you can spot the inequality. Generally we are given equalities to solve but this one is slightly different. The method remains the same throughout. You have to solve the inequality by factorization by grouping as it is a quadratic equation but the last step should be done correctly keeping in mind the inequalities.

Complete step by step solution:
 we will start the solution by writing the inequality given to us
 $ 2x + {x^2} \geqslant 35 $
We will try to first rewrite the equation
 $ {x^2} + 2x \geqslant 35 $
next step will be to subtract 35 from both sides of the equation which will be
 $
  {x^2} + 2x - 35 \geqslant 35 - 35 \\
   \Rightarrow {x^2} + 2x - 35 \geqslant 0 \;
  $
Now we have to use the method of factoring by grouping to get the factors. Hence the factorization will be as follows.
 $ {x^2} - 5x + 7x - 35 \geqslant 0 $
Taking the terms common from the two groups formed we get
 $ x\left( {x - 5} \right) + 7\left( {x - 5} \right) \geqslant 0 $
solving further we get
 $ \left( {x - 5} \right)\left( {x + 7} \right) \geqslant 0 $
so either $ \left( {x - 5} \right) \geqslant 0 $ or $ \left( {x + 7} \right) \geqslant 0 $
that is $ x \geqslant 5 $ or $ x \leqslant - 7 $ is the solution of the given inequality.
So, the correct answer is “ $ x \geqslant 5 $ or $ x \leqslant - 7 $ ”.

Note: While solving this type of sums if you are not sure of the inequalities that you have got as an answer you can always crosscheck it by substituting the values in the given main equation. If the inequality holds your answer is probably right. Also there are various rules of inequalities you need to keep in mind before solving