Question

What should be added to x^3 - 2x^2 + 3x + 1 to get 3x^2 + 9?​

Answer 1

Answer:

Given:-

What should be added to x³ - 2x² + 3x + 1 to get 3x² + 9?

To Find:-

The other number.

Note:-

●》Here, we will find the other number by; First number + Second number = Their sum.

●》For finding second number, we need to transpose first number and signs are also changed or not ( only in multiple and divisional value ). For example - Positive becomes Negative, Negative becomes Positive.

Solution:-

$\huge\red{First \ \ number = x³ - 2x² + 3x + 1}$

$\huge\red{Sum \ \ of \ \ first \ \ and \ \ second \ \ number = 3x² + 9}$

$\huge\red{\ \ \ \ Second \ \ number = ?}$

☆ According to note first point~

▪︎$First \ \ number + Second \ \ number = Their \ \ sum$

▪︎$x³ - 2x² + 3x + 1 + Second \ \ number = 3x² + 9$

☆ According to note second point ( transposing )~

▪︎$- 2x² + 3x + 1 + Second \ \ number = 3x² + 9 - x³$

▪︎$3x + 1 + Second \ \ number = 3x² + 9 - x³ + 2x²$

▪︎$1 + Second \ \ number = 3x² + 9 - x³ + 2x² - 3x$

▪︎$Second \ \ number = 3x² + 9 - x³ + 2x² - 3x - 1$

▪︎$Second \ \ number =$ 3x² + 9 - x³ + 2x² - 3x - 1

▪︎$Second \ \ number = 3x² + 2x² + 9 - 1 - x³ - 3x$

☆ After adding or subtracting like terms~

▪︎$Second \ \ number = 5x² + 8 - x³ - 3x$

$\huge\pink{Second \ \ number = 5x² + 8 - x³ - 3x}$

Checking:-

♤ Let's check that by adding first number to second gives their sum or not.

• $First \ \ number + Second \ \ number = Their \ \ sum \ \ ?$

• $x³ - 2x² + 3x + 1 + 5x² + 8 - x³ - 3x = 3x² + 9 \ \ ?$

• x³ - 2x² + 3x + 1 + 5x² + 8 - x³ - 3x = 3x² + 9 ?

• $x³ - x³ + 5x² - 2x² + 3x - 3x + 1 + 8 = 3x² + 9 \ \ ?$

• $0 + 3x² + 0 + 9 = 3x² + 9 \ \ ?$

• $3x² + 9 = 3x² + 9 \ \ ?$

♤ Yes!

$\huge\green{Hence, Proved : Second \ \ number = 5x² + 8 - x³ - 3x}$

Answer:-

Hence, the second number = 5x² + 8 - x³ - 3x.

:)