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

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 2

Solve the following equation:

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

Question 3

Solve the following equation:

$$ 2x^2-10x-12=0 $$

Question 4

$$ x^2+9=0 $$

Solve the equation

Question 5

Solve the following equation:

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

Examples with solutions for The Quadratic Formula

Exercise #1

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

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

Video Solution

Step-by-Step Solution

Let's recall the general form of the quadratic function:

$y=ax^2+bx+c$ The function given in the problem is:

$y=-x^2+25x$ $c$ is the free term (meaning the coefficient of the term with power 0),

In the function in the problem there is no free term,

Therefore, we can identify that:

$c=0$ Therefore, the correct answer is answer A.

Answer

$c=0$

Exercise #2

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 #3

Solve the following equation:

$2x^2-10x-12=0$

Video Solution

Step-by-Step Solution

Let's recall the quadratic formula:

$Quadratic formula | The formula$

We'll substitute the given data into the formula:

$x={{-(-10)\pm\sqrt{-10^2-4\cdot2\cdot(-12)}\over 2\cdot2}}$

Let's simplify and solve the part under the square root:

$x={{10\pm\sqrt{100+96}\over 4}}$

$x={{10\pm\sqrt{196}\over 4}}$

$x={{10\pm14\over 4}}$

Now we'll solve using both methods, once with the addition sign and once with the subtraction sign:

$x={{10+14\over 4}} = {24\over4}=6$

$x={{10-14\over 4}} = {-4\over4}=-1$

We've arrived at the solution: X=6,-1

Answer

$x_1=6$ $x_2=-1$

Exercise #4

$x^2+9=0$

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 #5

Solve the following equation:

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

Video Solution

Answer

$x_1=-1$ $x_2=-8$

Question 1

Solve the equation

$$ 3x^2-39x-90=0 $$

Question 2

Solve the following equation:

$$ -2x^2+22x-60=0 $$

Question 3

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

Identifies a,b,c

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

Question 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?

Question 5

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?

Exercise #6

Solve the equation

$3x^2-39x-90=0$

Video Solution

Answer

c = coefficient of the independent number

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

What are the components of the equation?

Video Solution

Answer

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

Question 1

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?

Question 2

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?

Question 3

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?

Question 4

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?

Question 5

a = coefficient of x²

b = coefficient of x

c = coefficient of the independent number

Answer

$c=5$

Topics learned in later sections