How do you find the discriminant and how many solutions does #1= -9x + 11x^2# have?

Question

Ethan Adkins · Accepted Answer

For a quadratic equation in standard form: #ax^2+bx+c=0#
the discriminant is #Delta = b^2-4ac# First rearrange the given equation into standard form
#1= -9x+11x^2# 11x^2-9x -1 =0# Evaluate the discriminant:
#Delta = (-9)^2 -4(11)(-1)#
#= 81+44 = 125# Since the solutions to a quadratic can be evaluated by the formula
#x=(-b+-sqrt(Delta))/(2a)#
It follows that 
#Delta { (< 0 rarr "no Real solutions"),(=0rarr "1 Real solution"),(>0rarr "2 Real solutions") :}# For this example, #Delta >0#
therefore there are 2 Real solutions

Ethan Adkins · Answer

undefined To find the discriminant of a quadratic equation $ ax^2 + bx + c = 0 $, where $ a \neq 0 $, the discriminant is calculated as $ b^2 - 4ac $.

For the equation $ 1 = -9x + 11x^2 $, rewrite it in the standard form $ ax^2 + bx + c = 0 $. So, $ 11x^2 - 9x + 1 = 0 $.

The coefficients are:
$ a = 11 $,
$ b = -9 $, and
$ c = 1 $.

Using the formula for the discriminant: $ b^2 - 4ac $:
$ (-9)^2 - 4(11)(1) = 81 - 44 = 37 $.

Since the discriminant is positive ($ 37 > 0 $), the quadratic equation $ 1 = -9x + 11x^2 $ has two real solutions.