The quadratic formula states:
For ax^2 + bx + c = 0, the values of x which are the solutions to the equation are given by:
x = (-b +- sqrt(b^2 - 4ac))/(2a)
The discriminate is the portion of the quadratic equation within the radical: color(blue)(b)^2 - 4color(red)(a)color(green)(c)
If the discriminate is:
- Positive, you will get two real solutions
- Zero you get just ONE solution
- Negative you get complex solutions
To find the value of b where there will be just ONE soluition, we set the discriminate equal to 0, substitute for a and c and solve for b:
Substitute:
color(red)(2) for color(red)(a)
color(blue)(-b) for color(blue)(b)
color(green)(-9) for color(green)(c)
color(blue)(-b)^2 - (4 * color(red)(2) * color(green)(-9)) = 0
color(blue)(-b)^2 - (-72) = 0
color(blue)(-b)^2 + 72 = 0
color(blue)(-b)^2 + 72 - color(red)(72) = 0 - color(red)(72)
color(blue)(-b)^2 + 0 = -72
color(blue)(-b)^2 = -72
color(red)(-1) * color(blue)(-b)^2 = color(red)(-1) * -72
b^2 = 72
sqrt(b^2) = +-sqrt(72)
b = +-sqrt(36 * 2)
b = +-sqrt(36)sqrt(2)
b = +-6sqrt(2)
