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 the Completing The Square Method.
This article will guide you to solving simple and complex quadratic equations using the Quadratic Formula. Relevant examples will enhance your understanding of the formula. Ensure that you read the article to the end.
What Is Quadratic Formula?
The quadratic formula, sometimes known as the almighty formula, is the most general method for solving equations of ax2 + bx + c = 0. Some quadratic equations that cannot be solved using factorization and completing the square methods can be efficiently solved 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, which you can denote by D.
Derivation Of The Quadratic Formula
Consider the general quadratic equation
ax2 + bx + c = 0
You then have to divide by using the coefficient of x2 (“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 must 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 discriminant value, D=0, the quadratic roots are real and equal. These 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
Knowledge of the nature of discriminants will help you solve quadratic equations using the quadratic formula method more quickly because you will have prior knowledge of how the quadratic roots will appear.
You can now use the quadratic formula to solve as many quadratic equations as possible. If you have any questions or comments regarding this article, please use the comment box below. Also, subscribe to this blog using your email to get more interesting educational guides.
