To solve a quadratic equation written in standard form ax2+bx+c=0, the Quadratic Formula can be used.
In this formula, the discriminant b2−4ac determines the number of real solutions of the quadratic equation.
2⋅2a=a
Commutative Property of Multiplication
a2+2ab+b2=(a+b)2
Commutative Property of Addition
(ba)m=bmam
(ab)m=ambm
ba=b⋅4aa⋅4a
Commutative Property of Multiplication
a⋅a=a2
Subtract fractions
ba=ba
a⋅b=a⋅b
a2=a
LHS−2ab=RHS−2ab
Put minus sign in numerator
Add and subtract fractions
In the Quadratic Formula, the expression b2−4ac, which is under the radical symbol, is called the discriminant.
A quadratic equation can have two, one, or no real solutions. Since the discriminant is under the radical symbol, its value determines the number of real solutions of a quadratic equation.
Value of the Discriminant | Number of Real Solutions |
---|---|
b2−4ac>0 | 2 |
b2−4ac=0 | 1 |
b2−4ac<0 | 0 |
Moreover, the discriminant determines the number of x-intercepts of the graph of the related quadratic function.
Use the Quadratic Formula: a=2,b=-4,c=10
-(-a)=a
Calculate power and product
Subtract term
-a=a⋅i
Calculate root
Simplify quotient