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An equation is a set of mathematical components related to each other by the mathematical operations. An essential condition for an equation is that the LHS and the RHS have to be equal. Only then can it be called an equation. Or else, it is termed inequality. Solving a quadratic equation is nothing but finding the roots of the given quadratic equation. The roots of the equation are the value of the variable (say ‘x’) in the equation. There are three different ways to solve a quadratic equation to find its roots. One among them is by using the quadratic formula.
Quadratic Equation
A quadratic equation is an equation with the highest degree of the equation being 2.
That is, the maximum exponential power on the variable is always ‘2’ in a quadratic equation.
A quadratic equation is mostly a polynomial. It contains more than one mathematical component, connected by the basic operations in maths.
A quadratic equation always has two roots as the value of the variable.
The most common format of a quadratic equation is ax² + bx + c.
In the quadratic equation, all the components are non-zero numbers.
Solving Quadratic Equation Using the Quadratic Formula
A quadratic equation is solved using three different methods:
Factoring
Using the quadratic formula
Completing the square
The Quadratic Formula
Using the quadratic formula is a direct method to identify the root values of a quadratic equation.
The quadratic formula is
x = -b √(b² – 4ac)
2a
To use the quadratic formula, the given quadratic equation should be in the format
ax² + bx + c = 0
If the given equation is not in the above-mentioned format, it should be simplified to that form before beginning the sum.
In the quadratic equation:
a is the coefficient of x² in the given equation.
b is the coefficient of x in the given equation.
c is the constant in the given equation.
The term (b² – 4ac) is called the discriminant (D) of the equation.
If the value of D is greater than zero, that is, if it is positive, the root values are real and unique.
If the value of D is lesser than zero, that is, if it is negative, the roots are imaginary.
If the value of D is equal to zero, the root values are one. That is, the values are equal.
Solving process – Using the Quadratic Formula
Steps for solving a quadratic equation using the quadratic formula are as follows. For example, consider the quadratic polynomial 5x² + 6x + 1.
The quadratic equation is verified to be in the format ax² + bx + c = 0.
The given equation is
5x² + 6x + 1
The quadratic equation formula is
x = -b √(b² – 4ac)
2a
From the given equation, a=5, b=6, c=1
Substituting the values in the quadratic equation formula:
x = -6 √[6² – 4(5)(1)]
2(5)
Simplifying:
x = -6 √[36 – 20]
10
x = -6 √16
10
x = -6 4
10
Calculating x values separately:
x = -6 + 4
10
x = -2/10
x = -0.2
x = -6 – 4
10
x = -10/10
x = -1
The roots of the given quadratic equation are x = -0.2 and -1.
Verification of the roots
To check if the root values are right, substitute them in the place of ‘x’ separately.
If the equation is solved to zero, then the values are correct.
5x² + 6x + 1
Substituting the root value x = -0.2 in the given quadratic equation:
= 5(-0.2)² + 6(-0.2) + 1
= 5(0.04) – 1.2 + 1
= 0.2 – 1.2 + 1
= 0.2 – 0.2
= 0
Substituting the root value x = -1 in the given quadratic equation:
= 5(-1)² + 6(-1) + 1
= 5(1) – 6 + 1
= 5 – 6 + 1
= 5 – 5
= 0
Since both the root values solve the equation to ‘0’, the obtained values are correct.
Solved Examples of Quadratic Equations by Using the Quadratic Formula
A few solved examples of the quadratic equations by using the quadratic formula are as follows:
x² + 5x + 6
The given equation is
x² + 5x + 6
The quadratic equation formula is
x = -b √(b² – 4ac)
2a
From the given equation, a=1, b=5, c=6
Substituting the values in the quadratic equation formula:
x = -5 √[5² – 4(1)(6)]
2(1)
Simplifying:
x = -5 √[25 – 24]
2
x = -5 √1
2
x = -5 1
2
Calculating x values separately:
x = -5 + 1
2
x = -4/2
x = -2
x = -5 – 1
2
x = -6/2
x = -3
The roots of the given quadratic equation are x = -2 and -3.
2x² + 9x – 5
The given equation is
2x² + 9x – 5
The quadratic equation formula is
x = -b √(b² – 4ac)
2a
From the given equation, a=2, b=9, c=-5
Substituting the values in the quadratic equation formula:
x = -9 √[9² – 4(2)(-5)]
2(2)
Simplifying,
x = -9 √[81 + 40]
4
x = -9 √121
4
x = -9 11
4
Calculating x values separately:
x = -9 + 11
4
x = 2/4
x = ½
x = -9 – 11
4
x = -20/4
x = -5
The roots of the given quadratic equation are x = 1/2 and -5.
Conclusion
A quadratic equation is an equation that has the highest exponential power on the variable as 2. The quadratic equation is solved to obtain the roots of the equation. These roots give the value of the variable in the equation. A quadratic equation can be solved using three different methods. Among these, using the quadratic formula is one of the straightforward methods to solve the given quadratic equation provided the equation is in the form ax² + bx + c = 0.
