factorise the following x²+6x+9 and x²+2x-15 and 49a²+70ab+25b² and 25x²-49y² and x²+6x+9
Answers
Answer:
If y = x^2 - 6x + 9, what is the value of x?
(1) y = 0
(2) x + y = 3
Step-by-step explanation:
1
We can Split the Middle Term of this expression to factorise it
In this technique, if we have to factorise an expression like ax2+b+c, we need to think of 2 numbers such that:
N1⋅N2=a⋅c=1⋅9=9
AND
N1+N2=b=6
After trying out a few numbers we get N1=3 and N2=3
3⋅3=9, and 3+3=6
x2+6x+9
=x2+3x+3x+9
=x(x+3)+3(x+3)
=(x+3)(x+3)
2
We can Split the Middle Term of this expression to factorise it.
In this technique, if we have to factorise an expression like ax2+bx+c, we need to think of 2 numbers such that:
N1⋅N2=a⋅c=1⋅(−15)=−15
AND
N1+N2=b=2
After trying out a few numbers we get N1=5 and N2=−3
5⋅(−3)=−15, and 5+(−3)=2
x2+2x−15=x2+5x−3x−15
=x(x+5)−3(x+5)
=(x−3)(x+5) .
(x−3)(x+5) is the factorised form of the expression.
3
The factors of 49a^{2}+70ab+25b^{2}49a2+70ab+25b2 is (7a+5b)and (7a+5b)
Solution:
Given 49a^{2}+70ab+25b^{2}49a2+70ab+25b2 ,
49=7^{2}49=72 and 25=5^{2}25=52
On expanding the equation we get,
\Rightarrow 7\times7\times a^{2}+70ab+ 5\times5\times b^{2}⇒7×7×a2+70ab+5×5×b2
\Rightarrow (7a)^{2}+70ab+(5b)^{2}⇒(7a)2+70ab+(5b)2
70ab can be written as 2\times5\times7\times ab2×5×7×ab
Therefore, \Rightarrow (7a)^{2}+2\times5\times7\times ab+(5b)^{2}⇒(7a)2+2×5×7×ab+(5b)2
It is in the form of a^{2}+2ab+b^{2} = (a+b)^{2}a2+2ab+b2=(a+b)2
Here, a is 7a and b is 5b
On substituting the values we get,
(7a\times5b)^{2}(7a×5b)2 or (7a\times5b)(7a\times5b)(7a×5b)(7a×5b)
Hence, the factors are (7a+5b)and (7a+5b).
4
25x^2-49y^2
5x^2-7y^2(a^2-b^2=(a-b)(a+b))
(5x-7y)(5x+7y)
5
We can Split the Middle Term of this expression to factorise it
In this technique, if we have to factorise an expression like ax2+b+c, we need to think of 2 numbers such that:
N1⋅N2=a⋅c=1⋅9=9
AND
N1+N2=b=6
After trying out a few numbers we get N1=3 and N2=3
3⋅3=9, and 3+3=6
x2+6x+9
=x2+3x+3x+9
=x(x+3)+3(x+3)
=(x+3)(x+3)
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