Is it correct to say: Applying the quadratic formula with a wrong sign of b?

-b uses the OPPOSITE sign of b. If b = -7, then -b = +7.

Is it correct to say: Concluding no roots exist when D = 0?

D = 0 means TWO equal real roots, not zero roots.

Is it correct to say: Ignoring 'positive/practical' constraint in word problems?

Age, length, width, speed must be positive — reject negative root with a one-line reason.

Is it correct to say: Dividing both sides by x and losing a root?

Never divide by a variable. Take terms to one side and factorise instead.

Is it correct to say: Using x = -b ± √(D)/2a without parentheses?

The entire numerator (-b ± √D) is divided by 2a — bracket carefully.

Is it correct to say: Trying to factorise when D is not a perfect square?

If D is not a perfect square, use the quadratic formula — factorisation will fail.

Quadratic Equations

Standard form, factorisation, quadratic formula, discriminant, nature of roots, applications — Maharashtra SSC Algebra

📐 10 formulas✏️ 3 examples🎯 6 practice⚖️ 6-8 marks🏫 Maharashtra State Board📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→One direct 3-mark quadratic formula problem is compulsory in the Algebra paper.
→Nature-of-roots question using discriminant D = b² − 4ac appears every year — memorise the three cases.
→Marathi terms: वर्गसमीकरण (quadratic equation), विवेचक (discriminant), मूळे (roots).
→Word problems: age, area, speed and time, geometry — always check whether both roots are physically valid (reject negative age, negative side, etc.).
→Show sum + product verification (α+β = -b/a, αβ = c/a) after solving — earns confirmation marks.

📐 Formulas(10)

Standard Form★ Board fav

a x^{2} + b x + c = 0, a \neq = 0

Symbol	Meaning
$a$	Coefficient of x² (must be non-zero)
$b$	Coefficient of x
$c$	Constant term

Quadratic Formula★ Board fav

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Discriminant★ Board fav

D = b^{2} - 4 a c

Sum of Roots

α + β = - \frac{b}{a}

Product of Roots

α β = \frac{c}{a}

Forming Quadratic from Roots

x^{2} - (α + β) x + α β = 0

Factorisation Method

a x^{2} + b x + c = a (x - α) (x - β)

Completing the Square

x^{2} + p x + q = 0 \Rightarrow (x + p /2)^{2} = (p /2)^{2} - q

Roots as Conjugate Surds

If one root is p + q, other is p - q

Relation Between Roots and Coefficients

α^{2} + β^{2} = (α + β)^{2} - 2 α β

✏️ Solved Examples

1Solved Exampleeasy6 steps

Solve 2x² − 7x + 3 = 0 using the quadratic formula. Verify with sum and product of roots.

Identify coefficients

a = 2, b = - 7, c = 3

2Solved Exampleboard4 steps

Find the values of k for which the equation kx² − 4x + 1 = 0 has equal roots.

Equal roots require D = 0

b^{2} - 4 a c = 0

3Solved ExampleHOTS6 steps

The sum of the squares of two consecutive positive integers is 365. Find the integers.

Let the integers be x and x + 1

x^{2} + (x + 1)^{2} = 365

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Applying the quadratic formula with a wrong sign of b
✓-b uses the OPPOSITE sign of b. If b = -7, then -b = +7.
2
Concluding no roots exist when D = 0
✓D = 0 means TWO equal real roots, not zero roots.
3
Ignoring 'positive/practical' constraint in word problems
✓Age, length, width, speed must be positive — reject negative root with a one-line reason.
4
Dividing both sides by x and losing a root
✓Never divide by a variable. Take terms to one side and factorise instead.
5
Using x = -b ± √(D)/2a without parentheses
✓The entire numerator (-b ± √D) is divided by 2a — bracket carefully.
6
Trying to factorise when D is not a perfect square
✓If D is not a perfect square, use the quadratic formula — factorisation will fail.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Solve x² − 5x + 6 = 0 by factorisation.
Q2
Find the discriminant of 3x² − 2x + 1 = 0 and state the nature of its roots.
Q3
Find k so that 2x² + kx + 8 = 0 has real and equal roots.
Q4
The product of two consecutive positive integers is 306. Find them.
Q5
The area of a rectangle is 60 m² and its length exceeds breadth by 4 m. Find dimensions.
Q6
If α and β are roots of x² − 6x + 5 = 0, find α² + β².

📝 Notes

Quadratic Equations

Quadratic equations describe everything from the arc of a thrown ball to the shape of a satellite dish. Class 10 focuses on solving them three ways — factorisation, completing the square, and the quadratic formula — and on interpreting when solutions are real, equal or complex.

SSC angle

Maharashtra Board Algebra reserves 6-8 marks for this chapter:

2-mark: solve by factorisation or state nature of roots.
3-mark: apply the quadratic formula, or find k for a given root condition.
4-mark HOTS: word problem — age, area, speed-time, integers.

Method-selection cheat sheet

Factorisation — try first. Works when D is a perfect square.
Completing the square — instructive; use when the problem explicitly asks for it.
Quadratic formula — universal, especially when D is not a perfect square.

Nature of roots — read D

| D = b² − 4ac | Nature | |---|---| | > 0 | Two distinct real roots | | = 0 | Two equal real roots (repeated) | | < 0 | No real roots (complex conjugates) |

Forming a quadratic from given roots α, β

x² − (α + β)x + αβ = 0

Word-problem checklist

Assign a variable and write the equation.
Reduce to ax² + bx + c = 0.
Solve.
Reject roots that violate the physical setting (negative age, side length, etc.).
State the final answer with units.

Board vs CBSE

Both boards teach identical formulas. Maharashtra SSC uses 'विवेचक' (discriminant) freely in Marathi papers and asks a fair share of Marathi-medium word problems on land, business and age.