An equation of the form ax2 + bx + c = 0, where a, b, c are real numbers and a0 is called a quadratic equation. The value of unknown variable x, which satisfies the given quadratic equation is called the roots of quadratic equation. For example, if is a root of quadratic equation ax2 + bx + c = 0, then a2 + b + c = 0. And the process of finding roots is known as solving a quadratic equation. There are three methods to solve a quadratic equation, which are as follows:

Solving a quadratic equation by factorisation.

Solving a quadratic equation by completing the square.

Solving a quadratic equation using quadratic formula.

In this article, we will learn about quadratic formula, its derivation and how to solve a quadratic equation using quadratic formula.

Quadratic formula is a formula that helps us to find the roots of a quadratic equation very easily by replacing the other methods of finding the roots like, factorisation method, completing the square method. The quadratic formula to find the roots of quadratic equation ax2 + bx + c = 0, where a ≠ 0 is given by:

\[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]

The plus (+) and minus (-) sign represents that quadratic formula will give two roots, one root corresponding to the plus (+) sign and another root corresponding to the minus (-) sign.

i.e., \[x_{1} = \frac{-b - \sqrt{b^{2} - 4ac}}{2a}\] and \[x_{2} = \frac{-b + \sqrt{b^{2} - 4ac}}{2a}\]. Both roots are evaluated by substituting the corresponding values of coefficients a, b and c from the quadratic equation ax2 + bx + c = 0.

Consider the quadratic equation ax2 + bx + c = 0, where a, b, c are real numbers and a0. Then,

ax2 + bx + c = 0

⇒ ax2 + bx = -c

On dividing both sides by a, we get:

⇒ \[x^{2} + \frac{b}{a}x = \frac{-c}{a}\]

Now adding \[(\frac{b}{2a})^{2}\] on both sides, we get:

⇒ \[x^{2} + \frac{b}{a}x + (\frac{b}{2a})^{2} = \frac{-c}{a} + (\frac{b}{2a})^{2}\]

⇒ \[(x + \frac{b}{2a})^{2} = (\frac{-c}{a} + \frac{b^{2}}{4a^{2}})\]

⇒ \[(x + \frac{b}{2a})^{2} = \frac{(b^{2} - 4ac)}{4a^{2}}\]

⇒ \[(x - \frac{b}{2a}) = \frac{\pm \sqrt{b^{2} - 4ac}}{2a}\], when (b2 - 4ac) 0

⇒ \[x = - \frac{b}{2a} \pm \frac{\sqrt{b^{2} - 4ac}}{2a}\]

⇒ \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\]

Therefore, the roots of quadratic equation ax2 + bx + c = 0 is \[x_{1} = \frac{-b + \sqrt{b^{2} - 4ac}}{2a}\] and \[x_{2} = \frac{-b - \sqrt{b^{2} - 4ac}}{2a}\].

The expression D = (b2 - 4ac) is called the discriminant of quadratic equation ax2 + bx + c = 0.

The roots of ax2 + bx + c = 0 are real only when D 0 i.e., b2 - 4ac 0 and the roots are given by \[x_{1} = \frac{-b + \sqrt{D}}{2a}\] and \[x_{2} = \frac{-b - \sqrt{D}}{2a}\].

To learn how to solve quadratic equations using quadratic formula, let us consider some examples and solve them using quadratic formula.

Example 1: Find the roots of quadratic equation 15x2 - x - 28 = 0 using quadratic formula.

Solution:

The given quadratic equation is 15x2 - x - 28 = 0. Comparing it with ax2 + bx + c = 0, we get a = 15, b = -1 and c = -28.

So, D = b2 - 4ac = (-1)2 - 4 × 15 × (-28) = 1681. As D = 1681 > 0, The given quadratic equation has real roots.

Now substituting the corresponding values of a, b and c in quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\], we get,

\[x = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4 \times 15 \times (-28)}}{2 \times 15}\]

⇒ \[x = \frac{1 \pm \sqrt{1681}}{30}\]

For plus (+) sign,

The root is \[x_{1} = \frac{1 + \sqrt{1681}}{30} = \frac{1 + 41}{30} = \frac{42}{30} = \frac{7}{5}\].

and, for minus (-) sign,

the root is \[x_{2} = \frac{1 - \sqrt{1681}}{30} = \frac{1 - 41}{30} = \frac{-40}{30} = \frac{-4}{3}\]

Hence, the required roots of quadratic equation 15x2 - x - 28 = 0 are 7/5 and -4/3 .

Example 2: Find the roots of quadratic equation x2 + 6x + 6 = 0 using quadratic formula.

Solution:

The given quadratic equation is x2 + 6x + 6 = 0. Comparing it with ax2 + bx + c = 0, we get a = 1, b = 6 and c = 6.

So, D = b2 - 4ac = (6)2 - 4 × 1 × 6 = 12. As D = 12 > 0, The given quadratic equation has real roots.

Now substituting the corresponding values of a, b and c in quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\], we get,

\[x = \frac{-(-6) \pm \sqrt{(6)^{2} - 4 \times 1 \times 6}}{2 \times 1}\]

⇒ \[x = \frac{-6 \pm \sqrt{12}}{2}\]

For plus (+) sign,

the root is \[x_{1} = \frac{-6 + \sqrt{12}}{2} = \frac{-6 + 2\sqrt{3}}{2} = (-3 + \sqrt{3})\]

and, for minus (-) sign,

the root is the root is \[x_{2} = \frac{-6 - \sqrt{12}}{2} = \frac{-6 - 2\sqrt{3}}{2} = (-3 - \sqrt{3})\]

Hence, the required roots of quadratic equation x2 + 6x + 6 = 0 are \[((-3 + \sqrt{3})\] and \[(-3 - \sqrt{3})\].

FAQ (Frequently Asked Questions)

Q.1) What's the Difference Between a Quadratic Equation and the Quadratic Formula?

Solution:

A quadratic equation is a polynomial equation having degree 2. The standard form of a quadratic equation in the variable x is given by: ax^{2} + bx + c = 0, where a, b and c are real numbers and a ≠ 0.

Whereas, the quadratic formula is a formula to determine the roots or solutions to the quadratic equation ax^{2} + bx + c = 0, which is given by:

x = [-b ± √(b^{2} - 4ac)]/2a

Also, the quadratic formula expresses the variable x in the quadratic equation ax^{2} + bx + c = 0, in terms of a, b and c.

Q.2) How is the Quadratic Formula Helpful for Finding the Roots of a Quadratic Equation?

Solution:

The quadratic formula for finding the roots of quadratic equation ax^{2} + bx + c = 0 is given by x = [-b ± √(b^{2} - 4ac)]/2a. It helps us to find the roots of a quadratic equation very easily by replacing the other methods of finding the roots like, factorisation method, completing the square method. The quadratic formula can be used for both factorable and non-factorable quadratic equations to find their roots in a simple manner.

Q.3) Why Should we Check the Value of Discriminant Before Using the Quadratic Formula?

Solution:

The discriminant of quadratic equation ax^{2} + bx + c = 0 is given by the expression D = (b^{2} - 4ac).

By checking the value of discriminant before using the quadratic formula, we can determine the nature of roots of the quadratic equation. So, if the value of D > 0, then the roots of quadratic equation will be Real and Unequal. And if the value of D = 0, then the roots of quadratic equation will be Real and Equal. While, if D < 0, then the roots are not real.