Question

1)Stiind ca a-b=4 si a²-b²=24,aflati a+b. 2)Stiind ca a+b=4 so a×b=2,calculati 1 supra a +1 supra b. 3)Stiind ca a+b=4 si a×b=2,calculati a²+b². 4)Stiind ca a-b=1,calculati a²+b²-2ab+2a-2b+1. 5)Stiind ca a+b=4 si ac+ad+bc+bd=4√3,calculati c+d.

Answer 1

$1).a-b=4 \\ a^2-b^2=24 \\ a+b=? \\ a^2-b^2=24 \\ (a-b)(a+b)=24 \\ 4(a+b)=24 \\ a+b=24:4 \\ \boxed{a+b=6}$
$\displaystyle 2).a+b=4 \\ a \cdot b=2 \\ \frac{1}{a} + \frac{1}{b} =? \\ \\ \frac{1}{a} + \frac{1}{b} = \frac{a+b}{a \cdot b} = \frac{4}{2} =2 \\ \\ \boxed{ \frac{1}{a} + \frac{1}{b} =2}$
$3).a+b=4 \\ a \cdot b=2 \\ a^2+b^2=? \\ a+b=4 \\ (a+b)^2=4^2 \\ a^2+2ab+b^2=16 \\ a^2+2 \cdot 2+b^2=16 \\ a^2+4+b^2=16 \\ a^2+b^2=16-4 \\ \boxed{a^2+b^2=12}$
$4).a-b=1 \\ a^2+b^2-2ab+2a-2b+1=? \\ a^2+b^2-2ab+2a-2b+1 =(a-b)^2+2(a-b)+1= \\ =1^2+2 \cdot 1+1=1+2+1=3+1=4 \\ \boxed{a^2+b^2-2ab+2a-2b+1=4}$
$5).a+b=4 \\ ac+ad+bc+bd=4 \sqrt{3} \\ c+d=? \\ ac+ad+bc+bd=4 \sqrt{3} \\ a(c+d)+b(c+d)=4 \sqrt{3} \\ (a+b)(c+d)=4 \sqrt{3} \\ 4(c+d)=4 \sqrt{3} \\ c+d=4 \sqrt{3} :4 \\ \boxed{c+d= \sqrt{3} }$