Quadratic Equations is an important mathematics topic for school exams, competitive exams, entrance tests and aptitude preparation. It is useful because many questions can be solved quickly when the definitions, formulas and examples are understood properly.
This tutorial explains quadratic equations in simple language. It includes meaning, important terms, formulas, step-by-step methods, shortcut ideas, solved examples, practice questions, answers, common mistakes and FAQs.
The best way to learn quadratic equations is to first understand the rule, then solve easy examples, and then try exam-style questions. This page follows the same order.
What Is Quadratic Equations?
A quadratic equation is a second-degree equation in which the highest power of the variable is 2. Its standard form is ax squared plus bx plus c equals 0, where a is not zero.
Important Terms
| Point | Details |
|---|---|
| Quadratic equation | Equation of degree 2 |
| Coefficient | Number multiplying a variable term |
| Constant | Fixed term |
| Root | Value satisfying the equation |
| Discriminant | b squared minus 4ac |
Important Formulas and Rules
| Point | Details |
|---|---|
| Standard form | ax^2 + bx + c = 0 |
| Quadratic formula | x = (-b plus or minus square root of b^2 – 4ac) / 2a |
| Discriminant | D = b^2 – 4ac |
| Sum of roots | -b/a |
| Product of roots | c/a |
Step-by-Step Method
- Write equation in standard form.
- Identify a, b and c.
- Try factorisation if simple.
- If not, use quadratic formula.
- Check roots by substitution.
Shortcut Ideas
- If product and sum are easy, factorise.
- D greater than 0 means two real roots.
- D equal to 0 means equal roots.
- D less than 0 means no real roots.
- Always keep a non-zero.
Common Question Types
| Point | Details |
|---|---|
| Factorisation | Split middle term |
| Formula method | Use quadratic formula |
| Nature of roots | Use discriminant |
| Form equation | Use sum and product |
| Word problem | Convert statement into equation |
Solved Examples
Example 1
Solve x^2 – 5x + 6 = 0.
x^2 – 5x + 6 = (x – 2)(x – 3). Roots are 2 and 3.
Example 2
Find discriminant of x^2 + 2x + 1.
D = 2^2 – 4 x 1 x 1 = 0.
Example 3
Find sum of roots of 2x^2 – 6x + 4.
Sum = -b/a = 6/2 = 3.
Example 4
Find product of roots of x^2 – 7x + 10.
Product = c/a = 10.
Example 5
Solve x^2 = 25.
x = 5 or -5.
Exam Style Examples
Question 1
Solve x^2 – 9x + 20 = 0.
Factors of 20 with sum 9 are 4 and 5. Roots are 4 and 5.
Use factorisation when numbers are simple.
Question 2
Nature of roots of x^2 + 4x + 4 = 0.
D = 16 – 16 = 0, so roots are real and equal.
Discriminant tells root nature.
Question 3
If roots are 2 and 3, form equation.
Equation is x^2 – 5x + 6 = 0.
Use sum and product of roots.
Common Mistakes
- Forgetting a cannot be zero.
- Wrong sign in -b/a.
- Not putting equation equal to zero.
- Making sign error in factorisation.
- Forgetting plus or minus in formula.
Practice Questions
- Solve x^2 – 4=0.
- Solve x^2 – 3x + 2=0.
- Find D for x^2 – 2x + 1.
- Sum roots of x^2 – 6x + 8.
- Product roots of x^2 – 6x + 8.
- Nature if D>0.
- Nature if D=0.
- Equation with roots 1 and 5.
- Solve x^2=16.
- a in 3x^2+2x+1.
Answers to Practice Questions
- 2 and -2
- 1 and 2
- 0
- 6
- 8
- Two real distinct roots
- Real equal roots
- x^2 – 6x + 5 = 0
- 4 and -4
- 3
Frequently Asked Questions
What is quadratic equation?
It is an equation with highest power 2.
What is standard form?
ax squared plus bx plus c equals 0.
What is discriminant?
It is b squared minus 4ac.
How many roots can a quadratic have?
It has two roots, real or complex.
When should formula be used?
Use it when factorisation is difficult.
Revision Notes
Quadratic equations are solved by factorisation, completing square or formula. Discriminant tells nature of roots.
For revision, keep a small formula list and solve a few mixed questions. Do not only read the solution. Try the question first, then compare your steps with the answer.
Conclusion
Quadratic equations become easy when standard form, factorisation and discriminant are clear.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.
Additional Exam Notes
Quadratic Equations questions should be solved with a clear setup. First identify the given values, then decide which formula or property is useful. If the expression has variables, write each step neatly so sign mistakes can be avoided.
Many students lose marks because they know the formula but apply it too quickly. Slow reading at the start saves time later. Check whether the question asks for value, root, degree, distance, slope, mean, median, mode, area or final conclusion.
After solving, verify the answer by substitution, estimation or unit checking. This is especially useful in algebra and coordinate geometry, where one wrong sign can change the whole answer.