Quadratic Formula - Examples, Exercises and Solutions

What is a quadratic equation?

Quadratic equations (also called second degree equations) contain three numbers called parameters:

  • Parameter a a represents the position of the vertex of the parabola on the Y Y axis. A parabola can have a maximum vertex, or a minimum vertex (depending on if the parabola opens upwards or downwards).
  • Parameter b b represents the position of the vertex of the parabola on the X X axis.
  • Parameter c c represents the point of intersection of the parabola with the Y Y axis.

These three parameters are expressed in quadratic equations as follows:

aX2+bX+c=0 aX^2+bX+c=0

They are called the coefficients of the equation.

So, how do we find the value of X X ?

To find X X and be able to solve the quadratic equation, all we need to do is to input the parameters (the number values of a, b and c) from the equation into the quadratic formula, and solve for X X .

For example:

3X2+8X+4=0 3X^2+8X+4=0

Practice Quadratic Formula

examples with solutions for quadratic formula

Exercise #1

Solve the following equation:

x2+5x+4=0 x^2+5x+4=0

Video Solution

Step-by-Step Solution

The parameters are expressed in the quadratic equation as follows:

aX2+bX+c=0

 

We substitute into the formula:

 

-5±√(5²-4*1*4) 
          2

 

-5±√(25-16)
         2

 

-5±√9
    2

 

-5±3
   2

 

The symbol ± means that we have to solve this part twice, once with a plus and a second time with a minus,

This is how we later get two results.

 

-5-3 = -8
-8/2 = -4

 

-5+3 = -2
-2/2 = -1

 

And thus we find out that X = -1, -4

Answer

x1=1 x_1=-1 x2=4 x_2=-4

Exercise #2

x2+9=0 x^2+9=0

Solve the equation

Video Solution

Step-by-Step Solution

The parameters are expressed in the quadratic equation as follows:

aX2+bX+c=0

 

We identify that we have:
a=1
b=0
c=9

 

We recall the root formula:

Roots formula | The version

We replace according to the formula:

-0 ± √(0²-4*1*9)

           2

 

We will focus on the part inside the square root (also called delta)

√(0-4*1*9)

√(0-36)

√-36

 

It is not possible to take the square root of a negative number.

And so the question has no solution.

Answer

No solution

Exercise #3

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number


what is the value of a a in the equation

y=3x10+5x2 y=3x-10+5x^2

Video Solution

Answer

a=5 a=5

Exercise #4

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


x2+4x5=0 x^2+4x-5=0

What are the components of the equation?

Video Solution

Answer

a=1 a=1 b=4 b=4 c=5 c=-5

Exercise #5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


10x2+5+20x=0 10x^2+5+20x=0

What are the components of the equation?

Video Solution

Answer

a=10 a=10 b=20 b=20 c=5 c=5

examples with solutions for quadratic formula

Exercise #1

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value of a a in the equation

y=x23x+1 y=-x^2-3x+1

Video Solution

Answer

a=1 a=-1

Exercise #2

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


56x2+12x=0 5-6x^2+12x=0

What are the components of the equation?

Video Solution

Answer

a=6 a=-6 b=12 b=12 c=5 c=5

Exercise #3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

Identifies a,b,c

5x2+6x8=0 5x^2+6x-8=0

Video Solution

Answer

a=5 a=5 b=6 b=6 c=8 c=-8

Exercise #4

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


8x25x+9=0 -8x^2-5x+9=0

What are the components of the equation?

Video Solution

Answer

a=8 a=-8 b=5 b=-5 c=9 c=9

Exercise #5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


x22=0 -x^2-2=0

What are the components of the equation?

Video Solution

Answer

a=1 a=-1 b=0 b=0 c=2 c=-2

examples with solutions for quadratic formula

Exercise #1

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


x2+7x=0 x^2+7x=0

What are the components of the equation?

Video Solution

Answer

a=1 a=1 b=7 b=7 c=0 c=0

Exercise #2

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value of b b in this quadratic equation:

y=4x216 y=4x^2-16

Video Solution

Answer

b=0 b=0

Exercise #3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value of b b in the equation

y=2x3x2+1 y=2x-3x^2+1

Video Solution

Answer

b=2 b=2

Exercise #4

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number


what is the value of b b in the equation

y=3x2+10x y=3x^2+10-x

Video Solution

Answer

b=1 b=-1

Exercise #5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number


what is the value of c c in this quadratic equation:

y=5+3x2 y=5+3x^2

Video Solution

Answer

c=5 c=5

Topics learned in later sections

  1. Methods for solving a quadratic function
  2. Squared Trinomial
  3. Completing the square in a quadratic equation