Understanding the Discriminant in a Quadratic Formula
By Kathleen Cantor, 03 Apr 2021
A quadratic equation in algebra is an equation in which the unknown variable's highest power is 2. You write quadratic equations using the following formula: ax² + bx + c = 0
Some quick examples of quadratic equations include:
2x² + 5x – 8 = 07x² + 9 = 0xx² – 26 = 3x
For this particular article, I will show you how the discriminant affects the solutions to quadratic equations. The discriminant of a quadratic formula is the part of the quadratic formula that determines the root type in a quadratic equation (imaginary, real, singular).
Solutions of a Quadratic Equation
Solutions to a quadratic equation are values of an unknown variable that make the equation true. There are four standard ways of finding the roots of a quadratic equation.
Factorization Method
This method is applicable if you can factorize the coefficients of the quadratic equation as av + bx + c = a( rx + n )( px + m ) = 0. Where n and m are the roots of the quadratic equation.
Square Method
This method is useful when you cannot factorize the coefficients of the quadratic equation as shown above. In completing the square method, the quadratic equation is expressed in the form
ax² + bx + c = x2 + ( b / a )x + ( c / a ) = 0x² + ( b / a )x + ( c / a ) = ( x +½b)2 + ( c / a ) – ( b² / 4 ) = 0( x +½b )2 = ( b² / 4 ) – ( c / a )
Solving for x gives the roots of the quadratic equation.
Quadratic Formula
You get the quadratic formula by completing the square method. If a quadratic equation is given as ax² + bx + c then the roots of the quadratic equation are given by x = (-b+-(b²–4ac )1/2 )/2a.
Graphical Method
In this method, you plot the quadratic equation, and the points where the graph cuts the x-axis are the roots of the equation.
For the purpose of this topic, however, we will focus on the quadratic formula.
The Discriminant of a Quadratic Formula
You can solve all quadratic equations using the quadratic formula method. Because of its versatility, we call it the almighty formula. You can find the roots of a quadratic equation using x = ( -b +- ( b² – 4ac )1/2 ) / 2a.
The term b² – 4ac under the square root determines the quadratic equation's roots and is the quadratic equation's discriminant. There are three possible outcomes for the discriminant.
b² - 4ac > 0
This happens when b² is greater than 4ac. If this is the case, you'll get two real roots of the quadratic equation. This is true because the square root of any positive number is a positive number. If you plot the graph of the quadratic equation, it will cut the x-axis at two points.
b² – 4ac = 0
This happens when b² is equal to 4ac. There is only one real root to the quadratic equation when this is your outcome. The square root of zero is zero. If you plot the quadratic equation graph, it will touch the x-axis at only one point.
(b² - 4ac) = 0
This happens when b² is less than 4ac. This is a job for imaginary roots. The roots are imaginary since the square root of a negative number is an imaginary number. The graph of such a quadratic equation will not touch the x-axis.
Let's illustrate the different cases where the discriminant determines the roots of quadratic equations.
Example 1
Find the following quadratic equations' roots:
x² + 7x + 3 = 03x² – 13x – 12 = 06y² + 10y = 0
Since we want to demonstrate how the discriminant affects a quadratic equation's roots, we will use the formula method to solve the problems above.
The quadratic formula is x = (-b + -(b² - 4ac)1 / 2) / 2a
Equation 1
x² + 7x + 3 = 0
a = 1, b = 7 and c = 3
Substitute the coefficient values of a, b, and c in the quadratic formula.
( -7 +- ( 72 – 4 * 1 * 3 )1/2 ) / ( 2 * 1 )
The discriminant here is ( 72 – 4 * 1 * 3 ) and comes to 37. Since 37 is greater than 0, it means that we have two real roots. Let us solve and get the roots!
( -7 +- ( 72 – 4 * 1 * 3 )1/2 ) / ( 2 * 1 )( -7 +- 371/2 ) / ( 2 * 1 )( -7 +- 6.08 ) / ( 2 * 1 )
The roots are
( -7 + 6.08 ) / ( 2 * 1 ) and ( -7 – 6.08 ) / ( 2 * 1 )-0.46 and -6.54
The roots of x² + 7x + 3 = 0 are -0.46 and -6.54
Equation 2
3x² – 13x – 12 = 0
a = 3, b = -13 and c = -12
After substituting the values for a, b and c in the formula, we have
( -(-13) +- ( -132 – 4 * 3 * -12 )1/2 ) / ( 2 * 3 )( 13 +- ( 313 )1/2 ) / ( 2 * 3 )( 13 +- 17.69 ) / ( 2 * 3 )
The roots are
( 13 + 17.69 ) / ( 2 * 3 ) and ( 13 – 17.69 ) / ( 2 * 3 )- 5.11 and -0.78
The roots of 3x2 – 13x – 12 = 0 are 5.11 and -0.78
Equation 3
6y² + 10y = 0
a = 6, b = 10 and c = 0
Coefficient c is zero, which is why it did not appear in the question.
After substituting the values for a, b, and c in the quadratic formula, we have this:
( -10 +- ( 102 – 4 * 6 * 0)1/2 ) / ( 2 * 6 )-
( -10 +- ( 102 )1/2 ) / ( 2 * 6 )
The roots are as follows:
( -10 + 10 ) / 12 and ( -10 – 10 ) / 120 and -1.67
For all the questions, the discriminant was greater than 0. All the roots are real and in pairs.
Example 2
Your quadratic equation is 2x2 + 4x + 2 = 0.
a = 2, b = 4 and c = 2
Substitute the coefficient values for a, b, and c in the quadratic equation.
( -4 +- ( 42 – 4 * 2 * 2 )1/2 ) / 2 * 2( -4 +- 0 ) / 4
The roots are
( -4 + 0 ) / 4 or ( -4 – 0 ) / 4- -1 and -1
The root of the quadratic equation is -1. In this example, the discriminant is equal to 0, and we arrived at only one root.
Example 3
Find the roots of 3x2 + 2x + 7 = 0.
a = 3, b = 2 and c = 7
Input your values for a, b, and c into the quadratic formula.
( -2 +- ( 22 – 4 * 3 * 7 )1/2 ) / 2 * 2( -2 +- ( -80 )1/2 ) / 4( -2 +- 8.9j ) / 4
The roots are
( -2 + 8.9j ) / 4 and ( -2 – 8.9j ) / 4- You cannot simplify the roots further.
The roots here are imaginary. They contain the imaginary variable j, which we define as ( -1 )1/2 or the square root of -1. We have arrived at imaginary roots because the discriminant was less than zero.
Wrapping Up
Understanding how the discriminant affects the outcome of your quadratic equation solutions is as easy as memorizing a formula. If you are ever faced with this mathematical task, always choose the quadratic formula.
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