This sheet covers the standard form of a quadratic equation, the two methods of finding its roots covered in this chapter, the quadratic formula, and the discriminant and what it tells you about the nature of the roots.
Because this chapter has only a handful of named formulas, it also lists the exact algebraic conditions you must check (such as \(a \neq 0\) and \(b^2 – 4ac \geq 0\)) that you will have to apply to exam questions.
Each formula below is grouped by topic, with the meaning and units of every symbol, when to use it, and worked examples with original numbers. For the derivations, the completing-the-square method, and full explanations, refer to the Quadratic Equations Class 10 notes.
Formulas at a Glance
| Purpose | Formula |
|---|---|
| Standard form of a quadratic equation | \( ax^2 + bx + c = 0, \ a \neq 0 \) |
| Root of a quadratic equation (definition) | \( a\alpha^2 + b\alpha + c = 0 \) |
| Roots by the quadratic formula, when \( b^2 – 4ac \geq 0 \) | \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \) |
| Two distinct real roots | \( b^2 – 4ac \gt 0 \) |
| Two equal real roots | \( b^2 – 4ac = 0 \) |
| No real roots | \( b^2 – 4ac \lt 0 \) |
| The repeated root when \( b^2 – 4ac = 0 \) | \( x = \frac{-b}{2a} \) |
All Formulas, Grouped by Topic
Quadratic Equations — Standard Form
A quadratic equation in the variable \( x \) is an equation of the form \[ ax^2 + bx + c = 0 \]
where \( a, \, b, \, c \) are real numbers and \( a \neq 0 \). Written this way — terms in descending order of degree — it is called the standard form of a quadratic equation (NCERT, p. 40). Note that \( a \neq 0 \) is essential: if \( a = 0 \), the equation becomes linear, not quadratic.
Any equation of the form \( p(x) = 0 \) where \( p(x) \) is a polynomial of degree 2 is a quadratic equation.
Roots of a Quadratic Equation
A real number \( \alpha \) is called a root (or solution) of the quadratic equation \( ax^2 + bx + c = 0, \( a \neq 0 \), if
Substituting \( x = \alpha \) into the equation makes the left-hand side equal to the right-hand side. A quadratic polynomial has at most two zeroes, so a quadratic equation can have at most two roots (NCERT, p. 43).
Solution by Factorisation
If the quadratic polynomial \( ax^2 + bx + c \) (with \( a \neq 0 \)) can be factorised into a product of two linear factors, say \( (px + q)(rx + s) \), then the equation \( ax^2 + bx + c = 0 \) is equivalent to The method of splitting the middle term is used to obtain the two linear factors; equating each factor to zero gives the roots (NCERT, p. 43).
There is no new formula here — this is revision of Class IX factorisation applied to quadratic equations.
Quadratic Formula
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are given by
When \( b^2 – 4ac \gt 0 \), the two roots are distinct:
\( \frac{-b}{2a} + \frac{\sqrt{b^2 – 4ac}}{2a} \) and \( \frac{-b}{2a} – \frac{\sqrt{b^2 – 4ac}}{2a} \).
When \( b^2 – 4ac = 0 \), both roots are \( \frac{-b}{2a} \), i.e. the equation has two equal real roots (NCERT, pp. 45-46).
Nature of Roots — The Discriminant
Since the expression \( b^2 – 4ac \) decides whether the equation has real roots, it is called the discriminant of the quadratic equation (NCERT, p. 46).
| Condition on discriminant | Nature of roots |
|---|---|
| \( b^2 – 4ac \gt 0 \) | Two distinct real roots |
| \( b^2 – 4ac = 0 \) | Two equal real roots (coincident roots) |
| \( b^2 – 4ac \lt 0 \) | No real roots |
This table is the list of the chapter’s summary points 5(i)-5(iii) (NCERT, p. 10). It follows from the quadratic formula: the square root of a positive number exists and gives two distinct values; the square root of zero is zero; and no real number squares to a negative number, so \( b^2 – 4ac \lt 0 \) gives no real roots (NCERT, p. 45).
What Each Symbol Means
| Symbol | What it means | Unit / nature of quantity |
|---|---|---|
| \( x \) | The unknown variable of the equation | Number (dimensionless) |
| \( a \) | Coefficient of \( x^2 \); must be non-zero | Real number |
| \( b \) | Coefficient of \( x \) | Real number |
| \( c \) | Constant term | Real number |
| \( \alpha \) | A real root of the equation; it satisfies \( a\alpha^2 + b\alpha + c = 0 \) | Real number |
| \( b^2 – 4ac \) | Discriminant — decides the nature of the roots | Real number (dimensionless) |
When to Use Each Formula
| Formula | When to use it |
|---|---|
| \( ax^2 + bx + c = 0 \) | To put any given equation into standard form first — before checking whether it is quadratic, before finding roots. Simplify the equation fully before deciding (some cubics reduce to quadratics). |
| \( a\alpha^2 + b\alpha + c = 0 \) | To check whether a proposed number \( \alpha \) is a root: substitute it and see whether the result is zero. |
| Factorisation method | When the quadratic trinomial factors neatly into linear factors (split the middle term). It is often the fastest method for simple integer cases. |
| Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \) | When factoring is not easy, or the roots are not rational. Only valid when \( b^2 – 4ac \geq 0 \); if the discriminant is negative, the equation has no real roots. |
| Discriminant tests | To determine whether an equation has two distinct real, equal real, or no real roots, and to solve questions like “Find \( k \) so that the equation has two equal roots” (set the discriminant equal to 0). |
Worked Examples
Example 1 — Applying the Quadratic Formula Directly
Find the roots of \( 2x^2 + 4x – 3 = 0 \).
Step 1: Identify the coefficients in the standard form \( ax^2 + bx + c = 0 \): \( a = 2 \), \( b = 4 \), \( c = -3 \).
\[ D = b^2 – 4ac = (4)^2 – 4(2)(-3) = 16 + 24 = 40 \]
Step 2: Since \( D \gt 0 \), apply the quadratic formula.
Final answer: The two distinct real roots are \( \frac{-2 + \sqrt{10}}{2} \approx 0.58 \) and \( \frac{-2 – \sqrt{10}}{2} \approx -2.58 \).
Example 2 — Using Discriminant to Check Feasibility
Is it possible to have a rectangle with perimeter 20 m and area 24 m²?
Step 1: Let the length be \( x \) m and the breadth be \( 10 – x \) m (since perimeter 20 means length + breadth = 10). Then the area condition gives:
Step 2: Check the discriminant: \( a = 1, b = -10, c = 24. \)
Step 3: Since \( D \gt 0 \), real dimensions exist. Apply the quadratic formula:
So \( x = 6 \) or \( 4 \).
Final answer: Yes, the rectangle is possible, with length 6 m and breadth 4 m (or vice versa).
Example 3 — Finding an Unknown Coefficient
Find the value of \( k \) such that \( kx^2 + 4x + 2 = 0 \) has two equal roots. (Note: since the roots must be equal, \( b^2 – 4ac = 0 \).)
Check: Let \( a = 2, \)\(, b = 4, \)\( c = 2. \) Then \( D = 16 – 16 = 0 \) as required.
Final answer: \( k = 2 \).
[Sign off: MathJax note: the *last* equation in Example 3 had a typo in writing “2x^2 + 4x + 2”.]
Common Mistakes to Avoid
| Mistake | Correct rule | How to check your answer |
|---|---|---|
| Using \( a = 0 \) in the quadratic formula (or treating it as quadratic) | The standard form requires \( a \neq 0 \). When \( a = 0 \), the equation is linear, not quadratic. |
Write the equation in the form \( ax^2 + bx + c = 0 \) and confirm the coefficient of \( x^2 \) is not zero before using any formula. |
| Forgetting the “ignored” root when it is negative in a geometry/word problem. | Length, breadth, distance, ages, and number of items must be positive. After finding roots, discard any negative value. | Substitute the remaining root back into the original problem (e.g., area = length \( \times \) breadth) and check whether it matches the given data. |
| Applying the formula when \( b^2 – 4ac \lt 0 \) | The quadratic formula is valid only when \( b^2 – 4ac \geq 0 \). If the discriminant is negative, the equation has no real roots. | Before computing, always compute the discriminant. A negative value tells you there are no real roots. |
Frequently Asked Questions
What is the difference between a polynomial and a quadratic equation?
A quadratic polynomial is an expression of the form \( ax^2 + bx + c \) with \( a \neq 0 \). A quadratic equation is the polynomial set equal to zero: \( ax^2 + bx + c = 0 \). The zeroes of the polynomial are exactly the same as the roots of the equation.
When should I use the factorisation method instead of the quadratic formula?
Use factorisation when the trinomial splits into linear factors easily — usually integer roots like the ones in the examples. Use the quadratic formula when factorisation requires fractions or roots that are not rational, or when you cannot spot the factors. The discriminant test first tells you whether a quadratic formula will even work.
I’m getting a negative value inside the square root. What does that mean?
That means the discriminant is negative. In this chapter, you conclude that the quadratic equation has no real roots. Common marks: write the discriminant value, state the condition \( b^2 – 4ac \lt 0 \), and then conclude the nature of the roots.
Final tip for revision: The formula list above is short — the discriminant and the quadratic formula solve the most exam problems. Practice writing the quadratic formula with signs correct in order to not lose marks.
NCERT solutions for Class 10 Maths Chapter 4: Quadratic Equations — practice the told textbook questions and apply the formulas yourself.
Reference: NCERT Class 10 Mathematics textbook, chapter Quadratic Equations.
Explore Class 10 Maths Formulas
- Previous: Pair of Linear Equations in Two Variables
- Next: Arithmetic Progressions
More for this chapter:
Related chapters:
- Real Numbers notes
- Polynomials notes
- Triangles notes
Official source: download the NCERT textbook free from ncert.nic.in.