Question

The sum of a whole number and twice the the sqaure of the number is 10. find the number

Answer 1

Solution:

$\begin{gathered}\\\end{gathered}$

Let the unknown number be x. Now, according to the question:

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{Number+2(Number^2) = 10}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{x+2(x^2) = 10}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{x+2x^2 = 10}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies{\boxed{\pink{\bf{x+2x^2-10 = 0}}}}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

Now, we arrived a quadratic equation. To solve this equation, we are going to use factorization method. Let us solve this equation with step by step explanation.

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2x^2+x-10 = 0}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

• x can be splitted as -4x+5x. This is known as splitting of the middle term.

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2x^2-4x+5x-10 = 0}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

• Now, let us take 2x and 5 as common factors.

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2x(x-2)+5(x-2) = 0}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

• Now, convert the equation in factorized form.

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{(2x+5)(x-2) = 0}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2x+5 = 0 \:or\:x-2 = 0}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2x = -5 \:or\:x = 2}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies{\boxed{\red{\bf{x = \dfrac{-5}{2}\:or\:2}}}}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

As we know that a whole number includes only positive numbers, negative and fractional numbers are rejected. Therefore:

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies{\boxed{\green{\frak{x = 2}}}}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

Hence, our required number is 2.

$\begin{gathered}\\\end{gathered}$

V E R I F I C A T I O N

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{x+2(x^2) = 10}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2+2(2^2) = 10}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2+2(4) = 10}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies\sf{2+8 = 10}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

$\begin{gathered}:\implies{\boxed{\orange{\frak{10 = 10}}}}\end{gathered}$

$\begin{gathered}\\\end{gathered}$

Hence, verified!