Quadratic Equations JEE Notes PDF | Free MCQs & Study Material Download

Omar Khayyam was famous during his lifetime as a mathematician, well known for inventing the method of solving equations by intersecting a parabola with a circle. Although his approach at achieving this had earlier been attempted by Menaechmus and others, Khayyám provided a generalization extending it to all cubics. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to Europe, where they contributed to the eventual development of non-Euclidean geometry.

In 1070 he wrote his greatest work on algebra. In it he classified equations according to their degree, and gave rules for solving quadratic equations, which are very similar to the ones in use today, and a geometric method for solving cubic equations with real roots. He also wrote on the triangular array of binomial coefficients known as Pascal's triangle.

In 1077, Omar wrote *Sharh ma ashkala min musadarat kitab Uqlidis* (Explanations of the Difficulties in the Postulates of Euclid). An important part of the book is concerned with Euclid's famous parallel postulate, which had also attracted the interest of Thabit ibn Qurra. Al-Haytham had previously attempted a demonstration of the postulate; Omar's attempt was a distinct advance.

Omar Khayyám also had other notable work in geometry, specifically on the theory of proportions.

**Introduction:** An equation of the form ax² + bx + c = 0 where a, b, c ∈ R and a ≠ 0 is called a quadratic equation, where a, b, c are called coefficients.

**Roots of Quadratic Equation:** To find roots of ax² + bx + c = 0 ⇒ x = [-b ± √(b² – 4ac)] / 2a. Sum of roots = –b/a, Product of roots = c/a. The equation whose roots are α, β is x² – (α+β)x + αβ = 0.

**Nature of Roots:** Discriminant D = b² – 4ac. If D < 0, roots are non-real complex conjugates. If D = 0, roots are real and equal = –b/(2a). If D > 0, roots are real and distinct. If D is a perfect square with rational a,b,c, roots are rational; if D is not a perfect square, roots are irrational conjugates. If a + b + c = 0 then 1 is a root; if a,b,c are rational then both roots are rational. If quadratic equation has more than two roots, then it is an identity: a = b = c = 0.

**Illustration:** Prove roots of (b–c)x² + 2(c–a)x + (a–b)=0 are real. Discriminant = [2(c–a)]² – 4(b–c)(a–b) = (a–b)²+(b–c)²+(c–a)² ≥ 0 ⇒ roots always real.

**Condition for Common Roots:** For a1x²+b1x+c1=0 and a2x²+b2x+c2=0, two common roots exist if a1/a2 = b1/b2 = c1/c2. For one common root α: α = (b1c2 – b2c1)/(a1b2 – a2b1) = (c1a2 – c2a1)/(a1b2 – a2b1).

**Quadratic Expression:** Expression ax²+bx+c is quadratic polynomial f(x)=ax²+bx+c.

**Graph:** y=ax²+bx+c is a parabola. If a>0 parabola opens upward; if a<0 downward. Vertex at (–b/(2a), –D/(4a)).

**Greatest/Least Value:** If a>0, minimum value = –D/(4a); if a<0, maximum value = –D/(4a).

**Sign of Quadratic Expression:** For a>0 & D<0, f(x)>0 for all x; for a<0 & D<0, f(x)<0 for all x; for a>0 & D>0, f(x)>0 outside roots α,β and f(x)<0 between roots; for a<0 opposite holds.

**Location of Roots:** Both roots < k if D≥0, a·f(k)>0, and (b/a)/2 > k; both roots > k if D≥0, a·f(k)>0, and (b/a)/2 < k; k between roots if D>0 and f(k)<0 when a>0 or f(k)>0 when a<0.

**Practice Exercises:**

1. If l,m,n real, l≠m, roots of (l–m)x²–5(l+m)x–2(l–m)=0 are (a) real equal (b) complex (c) real unequal (d) none.

2. If roots of x²–bx+c=0 are consecutive integers, b²–4c = (a)–2 (b)3 (c)2 (d)1.

3. α such that sum of squares of roots of x²–(α–2)x–α–1=0 is least: (a)1 (b)0 (c)3 (d)2.

4. If y=px+ a²p+1 is quadratic in p, equal roots if x²+y² = (a)–a² (b)0 (c)a² (d)none.

**Answers:** 1(c), 2(d), 3(a), 4(c).