# 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:

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

## examples with solutions for quadratic formula

### Exercise #1

Solve the following equation:

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

### 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

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

### Exercise #2

$x^2+9=0$

Solve the equation

### 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:

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.

No solution

### Exercise #3

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $a$ in the equation

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

### Video Solution

$a=5$

### Exercise #4

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

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

What are the components of the equation?

### Video Solution

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

### Exercise #5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

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

What are the components of the equation?

### Video Solution

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

## 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 $a$ in the equation

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

### Video Solution

$a=-1$

### Exercise #2

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

$5-6x^2+12x=0$

What are the components of the equation?

### Video Solution

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

### Exercise #3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

Identifies a,b,c

$5x^2+6x-8=0$

### Video Solution

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

### Exercise #4

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

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

What are the components of the equation?

### Video Solution

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

### Exercise #5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

$-x^2-2=0$

What are the components of the equation?

### Video Solution

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

## examples with solutions for quadratic formula

### Exercise #1

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

$x^2+7x=0$

What are the components of the equation?

### Video Solution

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

### Exercise #2

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

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

$y=4x^2-16$

### Video Solution

$b=0$

### Exercise #3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

what is the value of $b$ in the equation

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

### Video Solution

$b=2$

### Exercise #4

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $b$ in the equation

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

### Video Solution

$b=-1$

### Exercise #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:

$y=5+3x^2$

### Video Solution

$c=5$