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) = ax^{2} + bx + c (a ≠ 0) is called a quadratic expression.

→ Any equation of the form ax^{2} + bx + c = 0 (a ≠ 0) is called a quadratic equation.

→ Let the roots of the quadratic equation ax^{2} + 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 b^{2} – 4ac > 0 roots are real and distinct.

- b
^{2}– 4ac = 0 roots are equai and real, - b
^{2}– 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 ax^{2} + 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.

→ Let f(x) = ax^{2} + 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 ax^{2} + bx + c = 0 with α < β then

- for α < x < β, ax
^{2}+ bx + c and a have opposite signs. - for x < α or x > β, ax
^{2}+ bx + c and a have the same sign.

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