Quadratic formula: x = (−b ± √D) / 2a
The quadratic formula is x = (−b ± √D) / 2a, where D = b² − 4ac.
What each letter means
- \(a,\ b,\ c\) the numbers in ax² + bx + c = 0 (a ≠ 0)
- \(x\) the roots of the equation
- \(D=b^2-4ac\) the discriminant, which decides the nature of the roots
When to use it
To solve any quadratic equation ax² + bx + c = 0, especially one that does not factorise. Rearrange so that one side is 0 before you read off a, b and c.
Worked examples
1. Solve \(2x^2-7x+3=0\) using the quadratic formula.
- \(D=(-7)^2-4(2)(3)=49-24=25\)
- \(x=\dfrac{7\pm\sqrt{25}}{4}=\dfrac{7\pm5}{4}\)
Answer: \(x=3\) or \(x=\tfrac12\)
2. Find the nature of the roots of \(3x^2-2x+1=0\).
- \(D=(-2)^2-4(3)(1)=4-12\)
Answer: \(D=-8<0\), so there are no real roots
Common mistake
Losing a minus sign. If b = −3, then −b = 3 and b² = 9, not −9. Put negative values in brackets when you substitute.
On your course
| Course | In the exam |
|---|---|
| Class 10 | Learn it: CBSE gives no formula booklet |
From our own CBSE Maths formula sheets, in our words.
Practise and revise
- Practise Class 10 Quadratic Equations MCQs
- Revise Class 10 Quadratic Equations
- Print Class 10 one-page formula sheet
Questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √D) / 2a, where D = b² − 4ac. a, b, c: the numbers in ax² + bx + c = 0 (a ≠ 0); x: the roots of the equation; D = b²-4ac: the discriminant, which decides the nature of the roots.
Is the quadratic formula given in the exam?
Class 10: learn it: CBSE gives no formula booklet. This comes from our own CBSE Maths formula sheets; your teacher has the official booklet.
What does the discriminant tell you?
If b² − 4ac is positive there are two different real roots, if it is 0 there is one repeated root, and if it is negative there are no real roots.
Our own wording, examples and card, checked by CBSE Math Revision. Not produced or endorsed by CBSE or NCERT.