Quadratic Formula Explained with Examples for Class 10 and JEE

What Is the Quadratic Formula?

The quadratic formula is a ready-made recipe that gives the solutions of any quadratic equation. You do not have to guess factors or complete a square. You only put the three coefficients into the formula and calculate.

For an equation written as ax^2 + bx + c = 0, with a not equal to zero, the formula is:

x = (-b +/- sqrt(b^2 – 4ac)) / 2a

The sign +/- means “plus or minus”. It gives two answers, one with plus and one with minus. These answers are called the roots or solutions of the equation.

The formula is part of the NCERT Class 10 maths chapter on quadratic equations, and it stays useful in Class 11, in JEE algebra and in physics. If you want to check your work as you read, keep a free quadratic equation calculator open in another tab.

Meaning of Each Part of the Formula

Part	Name	What it does
a	Coefficient of x^2	Decides the shape of the curve. It can never be 0.
b	Coefficient of x	Shifts the curve sideways. Use its sign carefully.
c	Constant term	The point where the curve cuts the y-axis.
b^2 – 4ac	Discriminant (D)	Shows how many real roots there are.
2a	Denominator	Divides the whole top part, not just the root.

The most common slip is in the denominator. The 2a divides the entire expression -b +/- sqrt(D), so the fraction line must cover both parts.

Where the Formula Comes From

You do not need to memorise the proof, but seeing it once makes the formula feel less magical. Start with ax^2 + bx + c = 0 and divide by a: x^2 + (b/a)x + c/a = 0.

Move the constant: x^2 + (b/a)x = -c/a. Add (b/2a)^2 to both sides to complete the square: (x + b/2a)^2 = (b^2 – 4ac) / 4a^2. Take the square root of both sides: x + b/2a = +/- sqrt(b^2 – 4ac) / 2a. Finally, move b/2a across and you get the formula.

So the formula is nothing but completing the square done once for all equations. That is why it works every time.

The Discriminant and the Nature of Roots

Before using the full formula, calculate D = b^2 – 4ac. It tells you what to expect.

• If D is positive, there are two different real roots.
• If D is zero, there is one repeated real root, x = -b / 2a.
• If D is negative, there are no real roots, so you stop and write “no real roots”.

If D is a perfect square such as 25 or 1089, the roots are rational numbers, and the equation could also have been factorised. If D is not a perfect square, such as 12, the roots contain a square root and are irrational.

How to Use the Formula Step by Step

1. Rearrange the equation into ax^2 + bx + c = 0.
2. Write down the values of a, b and c with their signs.
3. Calculate the discriminant D = b^2 – 4ac.
4. If D is negative, stop. There are no real roots.
5. Find the square root of D.
6. Calculate x = (-b + sqrt(D)) / 2a and x = (-b – sqrt(D)) / 2a.
7. Simplify and check both roots in the original equation.

More Facts That Come from the Same Formula

The formula also gives two useful shortcuts. The sum of the roots is -b/a, and the product of the roots is c/a. The vertex of the curve y = ax^2 + bx + c lies at x = -b/2a, which is the middle of the two roots.

These facts help you check an answer without redoing the work, and they appear in many board and entrance questions.

Three Worked Examples with Indian Numbers

Example 1: 3x^2 – 7x + 2 = 0

Here a = 3, b = -7 and c = 2. Discriminant: D = (-7)^2 – 4(3)(2) = 49 – 24 = 25. The square root of 25 is 5.

x = (7 + 5) / 6 = 12 / 6 = 2, and x = (7 – 5) / 6 = 2 / 6 = 1/3. The roots are 2 and 1/3.

Check with sum and product: the sum is 2 + 1/3 = 7/3, which equals -b/a = 7/3. The product is 2/3, which equals c/a. Both match.

Example 2: x^2 – 4x + 1 = 0, irrational roots

Here a = 1, b = -4 and c = 1. D = 16 – 4 = 12. The square root of 12 is about 3.464, or exactly 2 sqrt(3).

x = (4 + 3.464) / 2 = 3.732 and x = (4 – 3.464) / 2 = 0.268. In exact form, the roots are 2 + sqrt(3) and 2 – sqrt(3). Because 12 is not a perfect square, factorising with whole numbers would never have worked here. This is where the formula shines.

Example 3: The train speed problem

A train covers 360 km at a uniform speed. If its speed had been 5 km/h more, it would have taken 1 hour less. Find the original speed. This is the type of question found in NCERT-style practice for Class 10.

Let the speed be x km/h. Time at speed x is 360/x hours. Time at speed x + 5 is 360/(x + 5) hours. The difference is 1 hour, so 360/x – 360/(x + 5) = 1.

Combine the fractions: 360(x + 5) – 360x = x(x + 5). The left side is 1,800, so x^2 + 5x – 1800 = 0. Here a = 1, b = 5, c = -1800.

D = 25 + 7,200 = 7,225, and its square root is 85. Then x = (-5 + 85) / 2 = 40, or x = (-5 – 85) / 2 = -45. A speed cannot be negative, so the original speed is 40 km/h.

Check: 360 / 40 = 9 hours, and 360 / 45 = 8 hours. The difference is exactly 1 hour.

Bonus: Finding k for equal roots

Board exams often ask: for what value of k does x^2 + kx + 9 = 0 have equal roots? Equal roots mean D = 0. So k^2 – 4(1)(9) = 0, giving k^2 = 36 and k = 6 or k = -6.

Key takeaway: Always write a, b and c with signs, find D first, and keep the 2a under the whole fraction line. These three habits remove most formula errors.

Benefits of the Quadratic Formula

• Works for every equation: No guessing and no special cases.
• Tells you the nature of roots early: The discriminant saves time on no-root questions.
• Helpful in word problems: Speed, area, age and profit questions all reduce to the same formula.
• Exam friendly: It is the standard method for CBSE boards, and it gives steady marks in JEE Main algebra.
• Foundation for graphs: The same numbers tell you where a parabola cuts the axis and where its turning point lies.

Challenges and Limitations

The formula is powerful but not always the fastest. For x^2 – 5x + 6 = 0, factorising takes ten seconds, while the formula needs several lines. Large coefficients also produce big discriminants, and squaring them by hand takes care.

The formula works only for equations with the highest power 2. Cubic or higher equations need other methods. It also gives roots as irrational numbers when D is not a perfect square, so you must decide how to round. For decimal arithmetic you can double-check with the decimal calculator.

Common Mistakes to Avoid

• Using the wrong sign for b: If b = -7, then -b = +7. Write it explicitly.
• Writing -b^2 instead of (-b)^2: Squaring a negative number gives a positive result, so (-7)^2 = 49.
• Dividing only the square root by 2a: The whole numerator must be divided by 2a.
• Forgetting the minus sign in -4ac: When c is negative, -4ac becomes positive, which is why D can grow large when c is negative.
• Using the formula before moving everything to one side: The right side must be zero first.
• Keeping a root that cannot exist: A negative speed or length should be rejected with a short reason.

Best Practices

• Write the formula once at the top of your working in an exam, then substitute.
• Compute b^2, 4ac and D on separate lines.
• Test whether D is a perfect square. If it is, your roots should be rational.
• Use the sum and product shortcut to verify the answers.
• Give irrational roots in surd form and also as decimals if the question asks.
• Practise with a few random equations, then check them in an online quadratic calculator.

Conclusion

The quadratic formula is a single line that solves every quadratic equation. Once you understand what a, b, c and the discriminant do, using it becomes calm and routine.

Work through the three examples above, including the train problem, and then try new ones of your own. With steady practice, quadratic questions will become some of the easiest marks in your maths paper.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = (-b +/- sqrt(b^2 – 4ac)) / 2a. It gives the solutions of ax^2 + bx + c = 0 for any values of a, b and c, as long as a is not zero.

What does +/- mean in the formula?

It means you calculate the answer twice: once with plus and once with minus. These give the two roots of the equation. If the square root part is zero, both answers are the same.

What if the number under the square root is negative?

Then the discriminant is negative and the equation has no real roots. In school maths you stop there and write that no real roots exist.

Why is a not allowed to be zero?

If a is zero, the x^2 term disappears and the equation becomes linear, such as bx + c = 0. Also the formula divides by 2a, and division by zero is not possible.

Can I use the formula when the equation is easy to factorise?

Yes. The formula always works, so it is a safe backup. For small numbers factorisation may be faster, but the formula gives the same roots.