How do you find the discriminant and how many and what type of solutions does #x^2 + 5x + 7 = 0# have?

2 Answers
Mar 10, 2018

Answer:

See a solution process below:

Explanation:

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 discriminant for this problem substitute:

#color(red)(1)# for #color(red)(a)#

#color(blue)(5)# for #color(blue)(b)#

#color(green)(7)# for #color(green)(c)#

Giving

#color(blue)(5)^2 - (4 * color(red)(1) * color(green)(7)) =>#

#25 - 28 =>#

#-3#

Because the number is negative you will get two complex solutions.

Mar 10, 2018

Answer:

#x^2+5x+7=0# has no real roots therefore no defined solutions.

Explanation:

D = # b^2 - 4ac #

A standard polynomial form is:
# ax^2 + bx + c = 0#

From your equation:
a= 1
b=5
c=7

∴ 25 - 4 x 1 x 7
= -3

A negative discriminant means there is no real root.
If you wanted to go further, it would have complex x roots. But yeah:

#x^2+5x+7=0# has no real roots therefore no defined solutions.