x is a natural number. (a) What number should be added to x 2+2x to get a perfect square ? 1 (b) If x 2+2x=15. Find the natural number represented by x.
Answers
Answer:
a) x² + 2x + 1
b) x = 3
Step-by-step explanation:
Step-by-step explanation:
x² + 2x + 1
b) x = 3
Step-by-step explanation:
\begin{gathered}a) \: \: \: {x}^{2} + 2x \\ Let \: \: us \: \: assume, \: \: x = 5 \\ {x}^{2} + 2x \\ = {5}^{2} + 2(5) \\ = 25 + 10 \\ = 35 \\ 36 \: \: is \: \: a \: \: perfect \: \: square \\ 1 \: \: should \: \: be \: \: added \: \: to \: \: the \: \: eqaution , \\ to \: \: get \: \: a \: \: perfect \: \: square \: \: as \: \: answer. \\ That \: \: is, \\ \large{ {x}^{2} + 2x + 1} \\ Let's \: \: check, \: \: once \: \: again \\ {x}^{2} + 2x + 1 \\ = {5}^{2} + 2(5) + 1 \\ = 25 + 10 + 1 \\ = 36 \: ( {6}^{2} ) \\ \\ (b) \: \: \: {x}^{2} + 2x = 15 \\ {x}^{2} + 2x - 15 = 0 \\ \: \: {x}^{2} + 5x - 3x - 15 = 0 \\ \: \: x(x + 5) - 3(x + 5) = 0 \\ \: \: (x + 5)(x - 3) = 0 \\ \: \: x + 5 = 0 \: \: \: or \: \: \: x - 3 = 0 \\ \: \: x = \underline{ \underline{ - 5}} \: \: \: or \: \: \: x = \underline{ \underline{3}} \\ \: \: \: \: \large{x = 3} \: (natural \: \: number) \end{gathered}
a)x
2
+2x
Letusassume,x=5
x
2
+2x
=5
2
+2(5)
=25+10
=35
36is a perfect square
1shouldbeaddedtotheeqaution,
to get a perfect square as answer.
ie.
x
2
+2x+1
Let
′
check,once again
x
2
+2x+1
=5
2
+2(5)+1
=25+10+1
=36(6
2
)
(b)x
2
+2x=15
x
2
+2x−15=0
x
2
+5x−3x−15=0
x(x+5)−3(x+5)=0
(x+5)(x−3)=0
x+5=0orx−3=0
x=
−5
orx=
3
x=3(naturalnumber)