Answer:
b > 36
Step-by-step explanation:
The general form of a quadratic equation is
[tex]ax^2 + bx + c = 0[/tex]
The roots of the quadratic formula above are given by
[tex]x_{1,2} = \dfrac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }[/tex]
The roots will be non-real whenever the term under the square root is negative since square root of a negative number is an imaginary number aka non-real
In other words,
b² - 4ac < 0 or
Instead of using b on the right side as given in the equation, I will use B to avoid confusion with the coefficient names of the standard equation
The given equation is
- x² + 8x+20= B
Move Bto the left side:
- x² + 8x +20 - B = 0
By comparing this equation to the general one we see
a = -1, b = 8, c = 20- B
The roots are imaginary when b² - 4ac < 0
b² - 4ac = 8² - ( 4 (-1) (20 - B) )
= 64 - (-4(20 - B))
= 64 - ( -80 + 4B)
= -64 + 80 - 4B
= 144 - 4B
If this is < 0 then|
144 - 4B <0
144 < 4B
or 4B > 144
Dividing by 4 both sides gives
B > 36
So whenever B > 36 the equation will have no real roots
