Property 5
\[ a\neq0 \qquad a,b,c\in\mathbb{R} \] \[ ax^2+bx+c=0 \]if the roots of the equation are \(\{k,n\}\) when they are put in the right place on the equation it will provide a correct equation.
\[ ak^2+bk+c=0 \] \[ an^2+bn+c=0 \]1.
\(x^2-ax-6=0\)
\(S.S.=\{-2,x_2\}\)
\(\Rightarrow a=?\)
2.
\(ax^2-5x+2=0\)
\(S.S.=\{2,x_2\}\)
\(\Rightarrow a=?\)
3.
\(3x^2-(a+1)x+4=0\)
\(S.S.=\{2,x_2\}\)
\(\Rightarrow a=?\)
4.
\(8x^2-ax+1=0\)
\(S.S.=\{-\samefrac12,x_2\}\)
\(\Rightarrow a=?\)
5.
\(3x^2-7x+a=0\)
\(S.S.=\{1,x_2\}\)
\(\Rightarrow a=?\)
6.
\(2x^2+ax-4=0\)
\(S.S.=\{-4,x_2\}\)
\(\Rightarrow x_2=?\)
7.
\(3x^2+(a+1)x-6=0\)
\(S.S.=\{-3,x_2\}\)
\(\Rightarrow x_2=?\)
Property 6
In radical expressions; the radical is placed in one side of the equation and then squares of both sides are taken. The roots which are found by this way are placed on the equation to control the accuracy.
1.
\(\sqrt{2x-1}=x\)
\(\Rightarrow S.S.=?\)
2.
\(\sqrt{4x+5}=x\)
\(\Rightarrow S.S.=?\)
3.
\(x=\sqrt{6x-8}\)
\(\Rightarrow S.S.=?\)
4.
\(\sqrt{2a+6}=a-1\)
\(\Rightarrow S.S.=?\)
5.
\(\sqrt{2x+17}-\sqrt{x}=3\)
\(\Rightarrow S.S.=?\)
6.
\(\sqrt{x+1}+\sqrt{x+3}=2\)
\(\Rightarrow x=?\)
7.
\(x-1=2\sqrt{x+1}+1\)
\(\Rightarrow x=?\)