The Quadratic Formula - Examples, Exercises and Solutions

What is a quadratic equation?

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

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

These three parameters are expressed in quadratic equations as follows:

$aX^2+bX+c=0$

They are called the coefficients of the equation.

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

To find $X$ 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 $X$ .

For example:

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

Practice The Quadratic Formula

Question 1

Solve the following equation:

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

Question 2

$$ x^2+9=0 $$

Solve the equation

Question 3

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

What is the value of $$ c $$ in the function $$ y=-x^2+25x $$ ?

Question 4

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

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

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

Question 5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

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

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

Examples with solutions for The Quadratic Formula

Exercise #1

Solve the following equation:

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

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

$x_1=-1$ $x_2=-4$

Exercise #2

$x^2+9=0$

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)

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

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ b $$ in the equation

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

Question 2

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

Question 3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

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

$$ y=-5x^2+4x-3 $$

Question 4

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of $$ b $$ in the equation

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

Question 5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

Answer

$a=-1$

Question 1

Solve the following:

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

Question 2

Solve the following equation:

$$ x^2+5x+6=0 $$

Question 3

Solve the following equation:

$$ x^2-3x+2=0 $$

Question 4

Solve the following equation:

$$ x^2-x-20=0 $$

Answer

$x_1=1,x_2=2$

Exercise #14

Solve the following equation:

$x^2-x-20=0$

Video Solution

Answer

$x_1=-4,x_2=5$

Exercise #15

Solve the following equation:

$x^2-4x+4=0$

Video Solution

Answer

$x=2$

Topics learned in later sections