Discriminant Practice AK

Using the discriminant, find out if the following equations have 1, 2, or no real solutions.  

1.

$\eqalign{x^2-7x+9=0\\b^2-4ac=D\\(-7)^2-4(1)(9)=D\\49-36=13\\D=13\\\text{There are two real solutions}}$

2.

$\eqalign{2x^2+4x-10=0\\b^2-4ac=D\\(4)^2-4(2)(-10)=D\\16+80=96\\D=96\\\text{There are two real solutions}}$

3.

$\eqalign{4x^2-x=-20\\4x^2-x+20=0\\b^2-4ac=D\\(-1)^2-4(4)(20)=D\\1-320=-319\\D=-319\\\text{There are no real solutions}}$

4.

$\eqalign{x^2=4x-8\\x^2-4x+8=0\\b^2-4ac=D\\(-4)^2-4(1)(8)=D\\16-32=-16\\D=-16\\\text{There are no real solutions}}$

5.

$\eqalign{x^2+6=-3x\\x^2+3x+6=0\\b^2-4ac=D\\(3)^2-4(1)(6)=D\\9-24=-15\\D=-15\\\text{There are no real solutions}}$

6.

$\eqalign{2x^2-6x+6=0\\b^2-4ac=D\\(-6)^2-4(2)(6)=D\\36-48=-16\\D=-12\\\text{There are no real solutions}}$

7.

$\eqalign{-6x+2=2x^2\\2x^2+6x-2=0\\b^2-4ac=D\\(6)^2-4(2)(-2)=D\\36+16=52\\D=52\\\text{There are two real solutions}}$

8.

$\eqalign{5x^2-2x+1=0\\b^2-4ac=D\\(-2)^2-4(5)(1)=D\\4-20=-16\\D=-16\\\text{There are no real solutions}}$

9.

$\eqalign{6x^2+6x+6=0\\b^2-4ac=D\\(-6)^2-4(6)(6)=D\\36-144=-108\\D=-108\\\text{There are no real solutions}}$

10.

$\eqalign{3x^2-12x=-12\\3x^2-12x-12=0\\b^2-4ac=D\\(-12)^2-4(3)(+12)=D\\144-144=0\\D=0\\\text{There is one real solution}}$