Quadratic Equations
Maharashtra State Board -
Mathematics Part 1
Complete Chapter Masterclass
1. What is a Quadratic Equation?
An equation in one variable where the maximum index (power) of the variable is 2.
Standard Form:
a\(x^2\) + b\(x\) + c = 0
- \(a, b,\) and \(c\) are real numbers.
- Crucial Rule: \(a \neq 0\). (If
\(a = 0\), the \(x^2\) disappears, and it becomes a linear equation!)
Practice: Standard Form
Q1: Decide whether the following are quadratic equations. If yes, write them in \(ax^2
+ bx + c = 0\) form and find \(a, b, c\).
1) \(y^2 = 5y - 10\)
2) \(x + \frac{1}{x} = -2\)
3) \((m+2)(m-5) = 0\)
2. Roots of a Quadratic Equation
The values of the variable which satisfy the equation (make LHS = RHS)
are called the Roots or Solutions of the equation.
How to check if a value is a root?
Substitute the given value in place of the variable (\(x, y, m\) etc.).
- If LHS = 0 (or RHS), then it is a root.
- If LHS \(\neq\) 0, then it is not a root.
Practice: Checking Roots
Q2: Determine whether the given values are roots of the quadratic equation:
\(x^2 + 4x - 5 = 0\), check for \(x = 1\) and \(x = -1\).
3. Factorization Method
Also known as "Splitting the Middle Term".
Steps to solve:
- Write the equation in \(ax^2 + bx + c = 0\) form.
- Find the product of \(a \times c\).
- Find two numbers whose product is \(ac\) and whose sum/difference
is \(b\) (the middle term).
- Split the middle term into these two parts.
- Take common factors from the first two and last two terms.
- Equate both brackets to zero to find roots.
Practice: Factorization
Q3: Solve the following quadratic equations by factorization:
1) \(x^2 - 15x + 54 = 0\)
2) \(2y^2 + 27y + 13 = 0\)
4. Completing the Square Method
Sometimes, equations cannot be easily factorized. We use this method to force the
equation into a perfect square format: \((x + k)^2\).
Magic Formula for the Third Term:
Third Term = \(\left( \frac{1}{2} \times \text{coefficient of } x \right)^2\)
Steps:
- Ensure \(a = 1\) (divide whole equation by \(a\) if it's not).
- Move constant \(c\) to the RHS.
- Calculate the Third Term using the formula.
- Add the Third Term to both sides (LHS & RHS).
- Write LHS as a perfect square and take the square root of both sides.
Practice: Completing the Square
Q4: Solve the following quadratic equation by completing the square method:
\(x^2 + 2x - 5 = 0\)
5. Formula Method
The ultimate tool that works on any quadratic equation!
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
How to use it safely to avoid calculation mistakes:
- Write standard form & compare to find \(a, b, c\).
- Calculate \(b^2 - 4ac\) separately first!
- Put the values into the main formula.
- Solve for the \(+\) case, then the \(-\) case.
Practice: Formula Method
Q5: Solve using the formula method:
\(5m^2 - 4m - 2 = 0\)
6. Nature of Roots (Discriminant)
The part of the formula under the square root, \(b^2 - 4ac\),
determines the nature of the roots. It is called the Discriminant (\(\Delta\)).
If \(\Delta = 0\)
Roots are Real and Equal.
If \(\Delta > 0\) (Positive)
Roots are Real and Unequal.
If \(\Delta < 0\) (Negative)
Roots are Not Real. (Because we can't take the square root of a negative
number in real math!).
Practice: Nature of Roots
Q6: Find the value of discriminant and determine the nature of roots for:
1) \(x^2 - 4x + 4 = 0\)
2) \(m^2 + 2m + 9 = 0\)
7. Relation between Roots & Coefficients
Let \(\alpha\) and \(\beta\) be the roots of \(ax^2 + bx + c = 0\).
Sum of roots:
\( \alpha + \beta = -\frac{b}{a}
\)
Product of roots:
\( \alpha \times \beta = \frac{c}{a}
\)
This means we can find the sum and product
without actually solving the equation!
Practice: Relation of Roots
Q7: For the quadratic equation \(10x^2 + 10x + 1 = 0\), find the values of:
1) \(\alpha + \beta\)
2) \(\alpha \times \beta\)
8. Forming an Equation from Roots
Reverse engineering! If we are given the roots \(\alpha\) and \(\beta\), how do we build
the original quadratic equation?
\( x^2 - (\alpha + \beta)x + \alpha\beta = 0 \)
In words:
\(x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0\)
Practice: Forming Equations
Q8: Obtain the quadratic equation if its roots are:
1) \(3\) and \(-10\)
2) \(2 - \sqrt{5}\) and \(2 + \sqrt{5}\)
9. Application of Quadratic Equations
How to tackle Word Problems:
- Read the problem carefully and identify the unknown quantity.
- Assign a variable (e.g., \(x\)) to the unknown.
- Translate the English conditions into Mathematical statements.
- Form the quadratic equation in terms of \(x\).
- Solve it using Factorization or the Formula method.
- CRITICAL: Interpret the roots! Reject unacceptable values (e.g., age cannot be
negative, length cannot be negative).
Practice: Word Problem (Numbers)
Q9: The sum of squares of two consecutive even natural numbers is 244. Find the
numbers.
Practice: Word Problem (Age)
Q10: The product of Pragati's age 2 years ago and 3 years hence is 84. Find her present
age.
Practice: Word Problem (Speed & Distance)
Q11: Pratik takes 8 hours to travel 36 km downstream and return to the same spot. The
speed of boat in still water is 12 km/hr. Find the speed of water current.
Practice: Word Problem (Geometry)
Q12: The hypotenuse of a right-angled triangle is 20 cm. The difference between its
other two sides is 4 cm. Find the lengths of the unknown sides.
Practice: Word Problem (Time & Work)
Q13: Suyash scored 10 marks more in the second test than that in the first. 5 times the
score of the second test is the same as square of the score in the first test. Find his score in the
first test.
Chapter Complete!
You have covered all concepts and methods of Quadratic Equations.