Quadratic Formula: Derivation, Proof & Solved Examples
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Quadratic Formula – Derivation, Proof & Examples (Standard Form)
Every quadratic formula problem starts from the standard form of a quadratic equation:
$ax^2 + bx + c = 0$
where $a$, $b$, and $c$ are real numbers and $a \neq 0$.
Here $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term. The condition $a \neq 0$ is essential — if $a = 0$, the equation becomes linear, not quadratic, and the quadratic formula no longer applies.
What Is the Quadratic Formula?
The quadratic formula, used to find the roots of any quadratic equation in standard form, is:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
This single quadratic equation formula gives both roots of the equation at once — one root using the $+$ sign and the other using the $-$ sign in front of the square root.
How Is the Quadratic Formula Derived? (Completing the Square Method)
The derivation of the quadratic formula starts from the standard form and uses the completing-the-square method:
Step 1: Start with $ax^2 + bx + c = 0$ and divide every term by $a$:
$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
Step 2: Move the constant term to the right side:
$x^2 + \frac{b}{a}x = -\frac{c}{a}$
Step 3: Add $\left(\frac{b}{2a}\right)^2$ to both sides to complete the square:
$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}$
Step 4: Write the left side as a perfect square:
$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$
Step 5: Take the square root of both sides:
$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$
Step 6: Isolate $x$ to get the final quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
This derivation is why the quadratic formula works for every quadratic equation, regardless of whether the equation can be factorised easily.
What Is the Discriminant in the Quadratic Equation Formula?
The expression under the square root in the quadratic formula, $D = b^2 - 4ac$, is called the discriminant, and it determines the nature of the roots without needing to solve the equation fully.
Nature of Roots Based on the Discriminant
| Condition | Nature of Roots |
|---|---|
| $D > 0$ | Two distinct real roots |
| $D = 0$ | Two equal real roots |
| $D < 0$ | No real roots (roots are complex/imaginary) |
Checking the discriminant first, before applying the full quadratic formula, is one of the fastest ways to predict the type of answer a question expects.
What Are the Sum and Product of Roots in the Quadratic Formula?
If $\alpha$ and $\beta$ are the two roots given by the quadratic formula for $ax^2 + bx + c = 0$, then Vieta's formulas give:
Sum of roots: $\alpha + \beta = -\frac{b}{a}$
Product of roots: $\alpha\beta = \frac{c}{a}$
These relations are often used to form a new quadratic equation directly from its roots, using $x^2 - (\alpha+\beta)x + \alpha\beta = 0$, without needing to re-derive the quadratic formula each time. These root relations are a recurring theme in JEE Main PYQ, so it's worth memorising both formulas cold.
When Should You Use the Quadratic Formula Instead of Factorisation?
The quadratic formula should be used when the equation cannot be factorised easily, especially when the roots are irrational (such as $2 \pm \sqrt{3}$) or complex. It always works for any quadratic equation and is the most reliable method under timed exam conditions, unlike factorisation, which only works quickly when the roots are simple integers.
Solved Examples Using the Quadratic Formula
Example 1: Solve $x^2 - 5x + 6 = 0$ using the quadratic formula.
Here $a = 1$, $b = -5$, $c = 6$
$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$
$x = 3$ or $x = 2$
Example 2: Solve $2x^2 + 3x - 2 = 0$ using the quadratic formula.
Here $a = 2$, $b = 3$, $c = -2$
$x = \frac{-3 \pm \sqrt{9 - 4(2)(-2)}}{4} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm 5}{4}$
$x = \frac{1}{2}$ or $x = -2$
Example 3: Find the nature of the roots of $x^2 - 4x + 4 = 0$ without solving fully.
$D = b^2 - 4ac = 16 - 16 = 0$
Since $D = 0$, the equation has two equal real roots.
Example 4: Solve $x^2 + 2 = 0$ using the quadratic formula.
Here $a = 1$, $b = 0$, $c = 2$
$x = \frac{0 \pm \sqrt{0 - 8}}{2} = \frac{\pm \sqrt{-8}}{2} = \pm i\sqrt{2}$
Since $D < 0$, the roots are complex.
Once you're comfortable with these solved examples, test yourself further using eSaral's JEE Test Series, which includes chapter-wise Algebra mock tests.
Explore More on eSaral (Related Quadratic Equations Resources)
Revise the complete Class 10 chapter: Quadratic Equation Class 10 Chapter 4 Short Notes →
Go deeper for JEE with the Class 11 chapter: Quadratic Equations Class 11 for IIT JEE →
Solve exercise-based questions step by step: NCERT Solutions for Class 10 Maths →
Download the complete syllabus reference: Class 10 Maths NCERT Book PDF →
Continue with Class 11 exercise solutions: NCERT Solutions for Class 11 Maths →
Watch it explained step by step: Quadratic Equations Revision Videos →
Practice quadratic-equation-based JEE Main questions with solutions: JEE Main Chapterwise PYQ with Solutions →
Confusing the quadratic formula with factorisation shortcuts under exam pressure? eSaral's Mathematics course, taught by IIT Bombay faculty, breaks down the quadratic formula derivation, discriminant tricks, and root relations with a 5-layer doubt-solving system so quadratic equations stay exam-ready every time. Start your free eSaral Mathematics demo class today →
Frequently Asked Questions
Find answers to common questions.
What is the discriminant in the quadratic formula?
The discriminant is the expression $D = b^2 - 4ac$ found under the square root in the quadratic formula, and it tells you the nature of the roots — real and distinct, real and equal, or complex — before solving the equation.
How is the quadratic formula derived?
The quadratic formula is derived from the standard form $ax^2 + bx + c = 0$ using the completing-the-square method, which rearranges the equation into a perfect square before isolating $x$.
What is the quadratic equation formula used for?
The quadratic equation formula is used to solve any quadratic equation for its roots, especially when the equation cannot be factorised easily or when the roots are irrational or complex.
What is the quadratic formula?
The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, used to find the roots of any quadratic equation written in the standard form $ax^2 + bx + c = 0$.
Does the quadratic formula always give real roots?
When should you use the quadratic formula instead of factorisation?
The quadratic formula should be used when the equation cannot be factorised easily, especially when the roots are irrational or complex, since factorisation only works quickly for simple integer roots.
What are the sum and product of roots in the quadratic formula?
For $ax^2 + bx + c = 0$, the sum of roots is $\alpha + \beta = -\frac{b}{a}$ and the product of roots is $\alpha\beta = \frac{c}{a}$, both derived directly from the quadratic formula.
What is the standard form of a quadratic equation?
The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.