`x (dx+10) =-3`
The equation above, where d is a constant, has no real solutions. The value of d could be which of the following?

Options

A. `-12`
B. 4
C. 8
D. 10

Answer

D

Explanation

Get the equation `x (dx +10) =-3`into the form `ax ^(2) + bx +c =0.` Multiply out the left side of the equation `x (dx + 10) =-3` to get `dx ^(2) + 10x =-3.` Add 3 to both sides to yield `dx ^(2) + 10x+3=0.`
When a,b, and c are all real, the equation `ax ^(2) + bx + c =0` (when `a ne 0)` does not have real solutions only if the discriminat, which is `b ^(2)-4 ac,` is negative. in the equation `dx ^(2) + 10x + 3=0, a=d, b =10 and c =3.` The discriminant, that is, `b ^(2)-4ac,` is negative. In the equation is `10 ^(2) -4 (d) (3) -100 -12d.`
Since you're looking for a negative discriminant, that is,` b ^(2)-4ac lt 0,` you need `100- 12 d lt 0.` Solve the inequality `100 -12d lt 0` for d:
`100- 12dlt 0`
`100 lt 12d`
`(100)/(12) lt d`
`(25)/(3) lt d`
`8 (1)/(3) lt d`
Among the answer choices, only 10 is greater than `8 (1)/(3),` so (D) is correct.