Question

Solve: \[{x^2} - 43x - 90 = 0\].

Answer 1

Hint: We are given with a quadratic equation of which roots are to be found. We can directly find the roots by the splitting method as the third term being their product and the second term being their sum.
Or can use the formula, \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].

Complete step by step answer:

Here given equation is a quadratic equation of the form\[a{x^2} + bx + c = 0\].
\[{x^2} - 43x - 90 = 0\]
Where the middle term is -43 and the last term is -90.
So the roots should be chosen in such a way that their sum should be -43 and the product should be -90.
Thus, we will list down the pair of numbers having a product is 90.
\[1 \times 90,2 \times 45,3 \times 30,5 \times 18,6 \times 15,9 \times 10\].
These are all pairs but we have to find a pair having sum -43. Thus 2 and 45 are the two numbers.
\[\begin{gathered}
  {x^2} - 43x - 90 = 0 \\
   \Rightarrow {x^2} - 45x + 2x - 90 = 0 \\
   \Rightarrow x(x - 45) + 2(x - 45) = 0 \\
   \Rightarrow \left( {x - 45} \right)\left( {x + 2} \right) = 0 \\
\end{gathered} \]
Separating the roots,
\[ \Rightarrow x - 45 = 0\] or \[x + 2 = 0\]
\[ \Rightarrow x = 45\] or \[x = - 2\]
This is the final answer for roots of above quadratic equation.

Note: Students if are unable to identify the roots directly then they should find the factors in pairs of the last term C and then decide the pair that fits for roots. Do take care of signs of the roots and terms because correct roots with wrong signs will help you in losing your marks only. We can also find the roots by using the formula \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].