The sum of a whole number and twice the the sqaure of the number is 10. find the number About the author Genesis
Solution: [tex]\begin{gathered}\\\end{gathered}[/tex] Let the unknown number be x. Now, according to the question: [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{Number+2(Number^2) = 10}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{x+2(x^2) = 10}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{x+2x^2 = 10}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies{\boxed{\pink{\bf{x+2x^2-10 = 0}}}}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] 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. [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2x^2+x-10 = 0}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] x can be splitted as -4x+5x. This is known as splitting of the middle term. [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2x^2-4x+5x-10 = 0}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] Now, let us take 2x and 5 as common factors. [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2x(x-2)+5(x-2) = 0}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] Now, convert the equation in factorized form. [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{(2x+5)(x-2) = 0}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2x+5 = 0 \:or\:x-2 = 0}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2x = -5 \:or\:x = 2}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies{\boxed{\red{\bf{x = \dfrac{-5}{2}\:or\:2}}}}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] As we know that a whole number includes only positive numbers, negative and fractional numbers are rejected. Therefore: [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies{\boxed{\green{\frak{x = 2}}}}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] Hence, our required number is 2. [tex]\begin{gathered}\\\end{gathered}[/tex] V E R I F I C A T I O N [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{x+2(x^2) = 10}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2+2(2^2) = 10}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2+2(4) = 10}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies\sf{2+8 = 10}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] [tex]\begin{gathered}:\implies{\boxed{\orange{\frak{10 = 10}}}}\end{gathered}[/tex] [tex]\begin{gathered}\\\end{gathered}[/tex] Hence, verified! Reply
Solution:
[tex]\begin{gathered}\\\end{gathered}[/tex]
Let the unknown number be x. Now, according to the question:
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{Number+2(Number^2) = 10}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{x+2(x^2) = 10}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{x+2x^2 = 10}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies{\boxed{\pink{\bf{x+2x^2-10 = 0}}}}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
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.
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2x^2+x-10 = 0}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2x^2-4x+5x-10 = 0}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2x(x-2)+5(x-2) = 0}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{(2x+5)(x-2) = 0}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2x+5 = 0 \:or\:x-2 = 0}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2x = -5 \:or\:x = 2}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies{\boxed{\red{\bf{x = \dfrac{-5}{2}\:or\:2}}}}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
As we know that a whole number includes only positive numbers, negative and fractional numbers are rejected. Therefore:
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies{\boxed{\green{\frak{x = 2}}}}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
Hence, our required number is 2.
[tex]\begin{gathered}\\\end{gathered}[/tex]
V E R I F I C A T I O N
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{x+2(x^2) = 10}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2+2(2^2) = 10}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2+2(4) = 10}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies\sf{2+8 = 10}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
[tex]\begin{gathered}:\implies{\boxed{\orange{\frak{10 = 10}}}}\end{gathered}[/tex]
[tex]\begin{gathered}\\\end{gathered}[/tex]
Hence, verified!