Solutions Using the Discriminant
Key Questions

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 
Answer:
To determine how many roots are there in a quadratic equation.
Explanation:
There are 4 natures
#b^24ac>0# and is a perfect square ># 2 rational roots#b^24ac>0# and is not a perfect square#># 2 irrational root
#b^24ac=0 ># 1 root
#b^24ac <0# No root 
Answer:
#Delta=b^24ac# for a quadratic#ax^2+bx+c=0# Explanation:
The discriminant indicated normally by
#Delta# , is a part of the quadratic formula used to solve second degree equations.
Given a second degree equation in the general form:
#ax^2+bx+c=0#
the discriminant is:
#Delta=b^24ac# The discriminant can be used to characterize the solutions of the equation as:
1)#Delta>0# two separate real solutions;
2)#Delta=0# two coincident real solutions (or one repeated root);
3)#Delta<0# no real solutions.For example:
#x^2x2=0#
Where:#a=1# ,#b=1# and#c=2#
So:
#Delta=b^24ac=1+4*2=9>0# , giving#2# real distinct solutions.The discriminant can also come in handy when attempting to factorize quadratics. If
#Delta# is a square number, then the quadratic will factorize, (since the square root in the quadratic formula will be rational). If it is not a square number, then the quadratic will not factorize. This can save you spending ages trying to factorize when it won't work. Instead, solve by completing the square or using the formula.I hope that helps!
Questions
Quadratic Equations and Functions

Quadratic Functions and Their Graphs

Vertical Shifts of Quadratic Functions

Use Graphs to Solve Quadratic Equations

Use Square Roots to Solve Quadratic Equations

Completing the Square

Vertex Form of a Quadratic Equation

Quadratic Formula

Comparing Methods for Solving Quadratics

Solutions Using the Discriminant

Linear, Exponential, and Quadratic Models

Applications of Function Models