Question

How do you solve $2{x^2} - 250x + 5000 = 0$?

Answer 1

Hint: First we identify the middle term of the given expression. Then we break the middle term into such numbers so that its product is equal to the product of the first and third term and those two numbers when added gives the original middle term. Once we have done that, we simply do grouping and take out the common factors and get our required factors for the sum.

Complete step-by-step solution:
The given expression is a quadratic function $2{x^2} - 250x + 5000 = 0$. It is a trinomial. A trinomial means an expression with three terms. We need to factorize this given expression to get our required expression. We can simplify this equation by taking out the common terms:
$2\left( {{x^2} - 125x + 2500} \right) = 0$
$ \Rightarrow {x^2} - 125x + 2500 = 0$
Factoring an expression simply means to find the roots of $x$ for which the equation gives the value as zero on solving. In order to do so, we first identify our middle term and only focus on the left hand side.
The middle term in the given expression is: $ - 125x$ , now we need to break the middle term into two such numbers so that it's equal to the product of the first and third term of the given expression and adds up to give the original middle term.
First term of the expression: ${x^2}$
Third term of the expression: $ + 2500$
Product of the first and third term: $2500{x^2}$
In order to break the middle term, into two numbers – we take the L.C.M of the product of the first and third term to get our required numbers:
$\begin{array}{*{20}{c}}
  {5\left| \!{\underline {\,
  {2500} \,}} \right. } \\
  {5\left| \!{\underline {\,
  {500} \,}} \right. } \\
  {5\left| \!{\underline {\,
  {100} \,}} \right. } \\
  {5\left| \!{\underline {\,
  {20} \,}} \right. } \\
  {2\left| \!{\underline {\,
  4 \,}} \right. } \\
  {2\left| \!{\underline {\,
  2 \,}} \right. } \\
  {\,\,1}
\end{array}$
The factors of the LCM are: $5 \times 5 \times 5 \times 5 \times 2 \times 2$
Now, we need to find two such numbers which when multiplied together gives $2500$. Thus, we take the two numbers as: $\left( {5 \times 5 \times 2 \times 2} \right)$and $\left( {5 \times 5} \right)$
Now, if we add $ - 100$ and$ - 25$ , we get $ - 125$
Thus we have our required two numbers; we just add the variable $x$ to them.
Thus our new expression becomes: ${x^2} - 100x - 25x + 2500$
Now we do further grouping by taking out the common factors:
$x\left( {x - 100} \right) - 25\left( {x - 100} \right)$
We see that $\left( {x - 100} \right)$ is a common factor in the above expression, hence it’s our first factor while the other factor is the common terms:

Thus the two factors are: $\left( {x - 100} \right)\left( {x - 25} \right)$

Note: Quadratic equations are the equations that are expressed in the form of $a{x^2} + bx + c$. If $a = 0$, then it is not a quadratic equation but a linear equation. We generally solve quadratic equations through factorization or grouping methods.