Property 3
As square root of a negative number are not real numbers;
\[ ax^2+bx+c=0 \]the roots of the equation change according to the symbol of \(\Delta\) as below:
- \(\Delta>0\): equation has two real roots. \[ x_1=\samefrac{-b+\sqrt{\Delta}}{2a}\qquad x_2=\samefrac{-b-\sqrt{\Delta}}{2a} \]
- \(\Delta=0\): The roots of \(\Delta=0\) equation are equal. \[ x_1=x_2=-\samefrac{b}{2a} \]
- \(\Delta<0\): There is no real roots for \(\Delta<0\) equation.
1.
\(ax^2-8x+16=0\)
\(S.S.=\{x_1,x_2\}\)
\(x_1=x_2\)
\(\Rightarrow a=?\)
2.
\(a\in\mathbb{R}^+\)
\(x^2+(a+1)x+36=0\)
\(S.S.=\{x_1,x_2\}\)
\(x_1=x_2\)
\(\Rightarrow a=?\)
3.
\(a\in\mathbb{R}^-\)
\(x^2-(a-2)x+4=0\)
\(S.S.=\{x_1,x_2\}\)
\(x_1=x_2\)
\(\Rightarrow a=?\)
4.
\(a\in\mathbb{R}^+\)
\(x^2-(a+2)x+2a+1=0\)
\(S.S.=\{x_1,x_2\}\)
\(x_1=x_2\)
\(\Rightarrow a=?\)
5.
\(x^2-10x+(a+6)=0\)
\(S.S.=\{x_1,x_2\}\)
\(x_1=x_2\)
\(\Rightarrow a=?\)
6.
\(x^2+6x+(a-2)=0\)
\(S.S.=\{x_1,x_2\}\)
\(x_1=x_2\)
\(\Rightarrow a=?\)
7.
\(x^2+8ax+16=0\)
\(S.S.=\emptyset\)
\(\Rightarrow ?
8.
\(x^2+4x+(a+1)=0\)
\(S.S.=\emptyset\)
\(\Rightarrow ?
9.
\(x^2-(a+1)x+9=0\)
\(S.S.=\emptyset\)
\(\Rightarrow ?
10.
\(ax^2+8x+1=0\)
If the equation has two different roots, which interval will “a” be located in?
11.
\((a+1)x^2-2x+3=0\)
If the equation has two different roots, which interval will “a” be located in?
12.
\(x^2-4x+a=0\)
If the equation has two different roots, which interval will “a” be located in?
13.
\(x^2-6x+(a+1)=0\)
If the equation has two different roots, which interval will “a” be located in?
14.
\(ax^2-4x+a=0\)
If the equation has two different roots, which interval will “a” be located in?