TS Inter 2nd Year Maths 2A Quadratic Expressions Formulas

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Learning these TS Inter 2nd Year Maths 2A Formulas Chapter 3 Quadratic Expressions will help students to solve mathematical problems quickly.

TS Inter 2nd Year Maths 2A Quadratic Expressions Formulas

→ A polynomial of the form f(x) = ax2 + bx + c (a ≠ 0) is called a quadratic expression.

→ Any equation of the form ax2 + bx + c = 0 (a ≠ 0) is called a quadratic equation.

→ Let the roots of the quadratic equation ax2 + bx + c = 0 be α, β then
α = \(\frac{-b+\sqrt{b^2-4 a c}}{2 a}\); β = \(\frac{-b-\sqrt{b^2-4 a c}}{2 a}\)
Now b2 – 4ac > 0 roots are real and distinct.

  • b2 – 4ac = 0 roots are equai and real,
  • b2 – 4ac < 0 roots are imaginary.
  • α + β = \(\frac{-b}{a}\): αβ = \(\frac{c}{a}\)

→ Let a, b, c be rational numbers, α, β are roots of the equation ax2 + bx + c = 0 then

  • α, β are equal rational numbers if Δ = 0.
  • α, β are distinct rational numbers if Δ is the square of a non-zero rational number.
  • α, β are conjugate surds if Δ > 0 and Δ is not the square of a rational number.

TS Inter 2nd Year Maths 2A Quadratic Expressions Formulas

→ Let f(x) = ax2 + bx + c = 0, a ≠ 0, α, β are roots of equation

  • if c ≠ 0 then αβ ≠ 0 and f(\(\frac{1}{x}\)) = 0 is an equation whose roots are \(\frac{1}{α}\) and \(\frac{1}{β}\)
  • f(x – k) = 0 is an equation whose roots are α + k and β + k.
  • f(- x) = 0 is an equation whose roots are – α and – β.

→ If α and β are roots of the equation ax2 + bx + c = 0 with α < β then

  • for α < x < β, ax2 + bx + c and a have opposite signs.
  • for x < α or x > β, ax2 + bx + c and a have the same sign.

→ If a < 0, the expression ax2 + bx + c has maximum at x = \(\frac{-\mathrm{b}}{2 \mathrm{a}}\) and the maximum value is given by \(\frac{4 a c-b^2}{4 a}\)

→ If a > 0, the expression ax2 + bx + c has minimum at x = \(\frac{-\mathrm{b}}{2 \mathrm{a}}\) and the minimum value is given by \(\frac{4 a c-b^2}{4 a}\)

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