Quadratic Equations – Complete Notes for Competitive Exams

1. Introduction

A quadratic equation is an equation of the form:$$ax^2 + bx + c = 0$$ax2+bx+c=0

where $a \neq 0$a=0, $b$b and $c$c are constants.

Quadratic equations are widely asked in Quantitative Aptitude exams.

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2. Standard Forms

• General form: $ax^2 + bx + c = 0$ax2+bx+c=0
• Factorized form: $(x – p)(x – q) = 0$(x−p)(x−q)=0
• Perfect square form: $(x + m)^2 = n$(x+m)2=n

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3. Methods to Solve Quadratic Equations

1. Factorization Method
   Split the middle term and factorize: $$x^2 + 5x + 6 = (x + 2)(x + 3) = 0$$x2+5x+6=(x+2)(x+3)=0
2. Quadratic Formula

$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$x=2a−b±b2−4ac​​

3. Completing the Square

$$x^2 + bx + c = 0 \implies (x + \frac{b}{2})^2 = \frac{b^2}{4} – c$$x2+bx+c=0⟹(x+2b​)2=4b2​−c

4. Cross Multiplication (if required in ratios)

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4. Discriminant (Δ)

$$\Delta = b^2 – 4ac$$Δ=b2−4ac

• Δ > 0 → Two real & distinct roots
• Δ = 0 → Two real & equal roots
• Δ < 0 → No real roots (complex roots)

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5. Sum and Product of Roots

For $ax^2 + bx + c = 0$ax2+bx+c=0

• Sum of roots: $\alpha + \beta = -\frac{b}{a}$α+β=−ab​
• Product of roots: $\alpha \beta = \frac{c}{a}$αβ=ac​

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6. Important Tips

• Always check factorization first
• Use discriminant to quickly determine root type
• Watch out for negative signs
• For competitive exams, practice mental calculation tricks

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Top 25 Practice Questions – Quadratic Equations

Q1.

Solve: $x^2 – 5x + 6 = 0$x2−5x+6=0

Q2.

Solve: $x^2 + 7x + 12 = 0$x2+7x+12=0

Q3.

Solve: $2x^2 + 5x – 3 = 0$2×2+5x−3=0

Q4.

Solve: $x^2 – 4x – 5 = 0$x2−4x−5=0

Q5.

Solve: $x^2 + 2x – 8 = 0$x2+2x−8=0

Q6.

Solve: $3x^2 – 8x + 4 = 0$3×2−8x+4=0

Q7.

Solve: $x^2 + x – 20 = 0$x2+x−20=0

Q8.

Solve: $x^2 – x – 30 = 0$x2−x−30=0

Q9.

Solve: $4x^2 – 12x + 9 = 0$4×2−12x+9=0

Q10.

Solve: $x^2 – 9 = 0$x2−9=0

Q11.

Solve: $2x^2 + 3x – 2 = 0$2×2+3x−2=0

Q12.

Solve: $x^2 – 10x + 25 = 0$x2−10x+25=0

Q13.

Solve: $x^2 + 5x + 4 = 0$x2+5x+4=0

Q14.

Solve: $x^2 – 6x + 8 = 0$x2−6x+8=0

Q15.

Solve: $3x^2 + 7x + 2 = 0$3×2+7x+2=0

Q16.

Solve: $x^2 – 2x – 35 = 0$x2−2x−35=0

Q17.

Solve: $x^2 + 4x + 3 = 0$x2+4x+3=0

Q18.

Solve: $2x^2 – 5x + 2 = 0$2×2−5x+2=0

Q19.

Solve: $x^2 – 16 = 0$x2−16=0

Q20.

Solve: $x^2 + 6x + 5 = 0$x2+6x+5=0

Q21.

Solve: $x^2 – x – 6 = 0$x2−x−6=0

Q22.

Solve: $x^2 + 9x + 18 = 0$x2+9x+18=0

Q23.

Solve: $x^2 – 7x + 10 = 0$x2−7x+10=0

Q24.

Solve: $x^2 – 8x + 15 = 0$x2−8x+15=0

Q25.

Solve: $x^2 + 3x – 18 = 0$x2+3x−18=0

Answer

Answers – Quadratic Equations

Q1. $x = 2, 3$x=2,3
Q2. $x = -3, -4$x=−3,−4
Q3. $x = \frac{1}{2}, -3$x=21​,−3
Q4. $x = 5, -1$x=5,−1
Q5. $x = 2, -4$x=2,−4
Q6. $x = 1, \frac{4}{3}$x=1,34​
Q7. $x = 4, -5$x=4,−5
Q8. $x = 6, -5$x=6,−5
Q9. $x = \frac{3}{2}$x=23​ (repeated root)
Q10. $x = 3, -3$x=3,−3
Q11. $x = \frac{1}{2}, -2$x=21​,−2
Q12. $x = 5$x=5 (repeated root)
Q13. $x = -1, -4$x=−1,−4
Q14. $x = 2, 4$x=2,4
Q15. $x = -\frac{1}{3}, -2$x=−31​,−2
Q16. $x = 7, -5$x=7,−5
Q17. $x = -1, -3$x=−1,−3
Q18. $x = \frac{1}{2}, 2$x=21​,2
Q19. $x = 4, -4$x=4,−4
Q20. $x = -1, -5$x=−1,−5
Q21. $x = 3, -2$x=3,−2
Q22. $x = -3, -6$x=−3,−6
Q23. $x = 2, 5$x=2,5
Q24. $x = 3, 5$x=3,5
Q25. $x = 3, -6$x=3,−6