Quadratic Formula ‖ Definition, Derivation And Examples
There are four (4) major ways of solving quadratic equations; Factorization, completing the squares, quadratic formula, and graphical methods. In our previous articles, we showed you how to solve quadratic equations using the Factorization method and Completing The Square Method.
This article will give you a step-by-step guide to solving simple and complex quadratic equations using the Quadratic Formula. We shall enhance your understanding of the quadratic formula by using relevant examples to describe it. Ensure to read the article to the end.
What Is Quadratic Formula?
The quadratic formula, sometimes known as the almighty formula, is the most general method used in solving equations of the form ax2 + bx + c = 0. Some quadratic equations which cannot be solved using factorization and completing the square methods can be solved easily using the quadratic formula.
Mathematically, the quadratic formula is given as:
Where:
x = roots of the quadratic equation
b2 – 4ac = discriminant of the quadratic equation and you can denote it by D.
Derivation Of The Quadratic Formula
Consider the general quadratic equation
ax2 + bx + c = 0
You then have to divide through with the coefficient of x2 (which is “a” in the equation) to make x2 unity.
ax2/ a + bx/a + c/a = 0
Next, you transfer the constant term (the term without an unknown) to the right-hand side of the equation
x2 + bx/a = -c/a
Make the Left-Hand Side of the equation a perfect square. To do this, you have to add the square of half the coefficient of x to both sides of the equation
bx/a = (b/2a)2
x2 + bx/a + (b/2a)2 = – c/a + (b/2a)2
Simplifying the equation
x2 + bx/a + b2/4a2 = – c/a + b2/4a2
x2 + bx/a + b2/4a2 = – 4ac + b2 / 4a2
Next, you factorize the left-hand side of the equation
(x + b/2a)2 = b2 – 4ac / 4a2
To remove the squares, take the square root of both sides
Now you can see how the quadratic formula was derived. In solving equations using the quadratic formula, you have to take special care of the negative signs. It can alter your results if you are not careful enough with the signs. However, if you gain mastery of this formula, you can solve any quadratic equation no matter how complicated it may be.
Worked Example
Using the quadratic formula, solve the quadratic equation 2x2 + 15x + 7 = 0
Solution
The quadratic equation 2x2 + 15x + 7 = 0 can be compared with the general quadratic equation ax2 + bx + c = 0. From here, you can deduce that:
a = 2
b = 15 and
c = 7
See Also:
Quadratic Equation: How To Solve Using Factorization Method
Simultaneous Equation- Meaning, Methods, and Examples
Nature Of The Quadratic Discriminant
You can recall that b2 – 4ac is defined as the discriminant of the quadratic equation denoted by D. The character of the roots of the quadratic equation depends on the value of the discriminant, D. The characteristics or nature of D is given as follows:
-
When D is positive:
That is the value of D is greater than 0, then the two roots of the quadratic equation are real and different.
For example, consider a quadratic equation x2 -2x – 4 = 0, using the general quadratic formula, a = 1, b = -2 and c = -4
The discriminant b2 – 4ac will be given as:
(-2)2 – 4 (1)(-4) = 4 + 16
= 20
Now D>0 therefore the roots of the quadratic equation are sure to be real and different numbers.
-
When the value of D is negative:
That is the value of D is less than 0, the roots are said to be imaginary or complex. They are not real numbers.
Consider a quadratic equation x2 -2x + 4 = 0, using the general quadratic formula, a = 1, b = -2 and c = 4
The discriminant b2 – 4ac will be given as:
(-2)2 – 4 (1)(4) = 4 – 16
= -12
The roots of the quadratic equation will be imaginary since the square root of a negative number is not real.
-
When D = 0:
When the value of the discriminant, D= 0, the quadratic roots are real and equal. The quadratic roots are said to be Coincident roots.
When D = 0, √b2 – 4ac will be = 0 and the roots will be given by:
x = – b/2a
Consider a quadratic equation x2 -2x + 1= 0, using the general quadratic formula, a = 1, b = -2 and c = 1
The discriminant b2 – 4ac will be given as:
(-2)2 – 4 (1)(1) = 4 – 4
= 0
The roots of the quadratic equation will be:
x = – b/2a
= – (-2)/(2 x 1) = 2/2
X = 1 twice
The knowledge of the nature of discriminant will help you solve quadratic equations using the quadratic formula method easier because you will have prior knowledge of how the quadratic roots will appear.
You can now go ahead to solve as many quadratic equations as possible using the quadratic formula. If you have any questions or comments regarding this article, endeavor to use the comment box below. Also, subscribe to this blog using your email to get more interesting educational guides.
