Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 3 Quadratic Expressions to solve questions creatively.

## Intermediate 2nd Year Maths 2A Quadratic Expressions Formulas

→ If a, b, c are real or complex numbers and a ≠ 0, then the expression ax^{2} + bx + c is called a quadratic expression in the variable x.

Eg: 4x^{2} – 2x + 3

→ If a, b, c are real or complex numbers and a ≠ 0, then ax^{2} + bx + c = 0 is called a quadratic equation in x.

Eg: 2x^{2} – 5x + 6 = 0

→ A complex number α is said to be a root or solution of the quadratic equation ax^{2} + bx + c = 0 if aα^{2} + bα + c = 0

→ The roots of ax^{2} + bx + c = 0 are \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

→ If α, β are roots of ax^{2} + bx + c = 0, then α + β = \(\frac{-b}{a}\) and αβ = \(\frac{C}{a}\)

→ The equation of whose roots are α, β is x^{2} – (α + β)x + αβ = 0

→ Nature of the roots: ∆ = b^{2} – 4ac is called the discriminant of the quadratic equation ax^{2} + bx + c = 0. Let α, β be the roots of the quadratic equation ax^{2} + bx + c = 0

Case 1: If a, b, c are real numbers, then

- ∆ = 0 ⇔ α = β = \(\frac{-b}{2 a}\) (a repeated root or double root)
- ∆ > 0 ⇔ α and β are real and distinct.
- ∆ < 0, ⇔ α and β are non- real complex numbers conjugate to each other.

Case 2: If a, b, c are rational numbers, then

- ∆ = 0 ⇔ α and β are rational and equal (ei) α = \(\frac{-b}{2 a}\), a double root or a repeated root.
- ∆ > 0 and is a square of a rational number ⇔ α and β are rational and distinct.
- ∆ > 0 but not a square of a rational number ⇔ α and β are conjugate surds.
- ∆ < 0, ⇔ α and β are non- real ⇔ α and β are non-real con conjugate complex numbers.

→ Let a, b and c are rational numbers, α and β be the roots of the equations ax^{2} + bx + c = 0. Then

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

→ If a_{1}x^{2} + b_{1}x + c_{1} = 0, a_{2}x^{2} + b_{2}x + c_{2} = 0 have two same roots, then \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

→ If α, β are roots of ax^{2} + bx + c = 0,

- the equation whose roots are \(\frac{1}{\alpha}, \frac{1}{\beta}\) is f \(\left(\frac{1}{x}\right)\) = 0. If c ≠ 0 (ie) αβ ≠ 0
- the equation whose roots are α + k, β + k is f(x – k) = 0
- the equation whose roots are kα and kβ is f\(\left(\frac{x}{k}\right)\) = 0
- the equation whose roots are equal but opposite in sign is f(-x) = 0

(ie) the equation whose roots are – α, – β is f(-x) = 0.

→ If the roots of ax^{2} + bx + c = 0 are complex roots then for x ∈ R, ax^{2} + bx + c and ‘a’ have the same sign.

→ If α and β (α < β) are the roots of ax^{2} + bx + c = 0 then

- ax
^{2}+ bx + c and ‘a’ are of opposite sign when α < x < β - ax
^{2}+ bx + c and ‘a’ are of the same sign if x < α or x > β.

→ Let f(x) = ax^{2} + bx + c be a quadratic function

- If a > 0 then f(x) has minimum value at x = \(\frac{-b}{2 a}\) and the minimum value is given by \(\frac{4 a c-b^{2}}{4 a}\)
- If a < 0 then f(x) has maximum value at x = \(\frac{-b}{2 a}\) and the maximum value is given by \(\frac{4 a c-b^{2}}{4 a}\)

→ A necessary and sufficient condition for the quadratic equation a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0 to have a common root is (c_{1}a_{2} – c_{2}a_{1})^{2} = (a_{1}b_{2} – a_{2}b_{1}) (b_{1}c_{2} – b_{2}c_{1}).

→ If a_{1}b_{2} – a_{2}b_{1} = 0 then common root of a_{1}x^{2} + b_{1}x + c_{1} = 0, a_{2}x^{2} + b_{2}x + c_{2} = 0 is \(\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\).

→ The standard form of a quadratic ax^{2} + bx + c = 0 where a, b, c ∈ R and a ≠ 0

→ The roots of ax^{2} + bx + c = 0 are \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)

→ For the equation ax^{2} + bx + c = 0, sum of the roots = \(-\frac{b}{a}\), product of the roots = \(\frac{c}{a}\).

→ If the roots of a quadratic are known, the equation is x^{2} – (sum of the roots)x +(product of the roots)= 0

→ “Irrational roots” of a quadratic equation with “rational coefficients” occur in conjugate pairs. If p + √q is a root of ax^{2} + bx + c = 0, then p – √q is also a root of the equation.

→ “Imaginary” or “Complex Roots” of a quadratic equation with “real coefficients” occur in conjugate pairs. If p + iq is a root of ax^{2} + bx + c = 0. Then p – iq is also a root of the equation.

→ Nature of the roots of ax^{2} + bx + c = 0

Nature of the Roots | Condition |

Imagine | b^{2} – 4ac < 0 |

Equal | b^{2} – 4ac = 0 |

Real | b^{2} – 4ac ≥ 0 |

Real and different | b^{2} – 4ac > 0 |

Rational | b^{2} – 4ac is a perfect square a, b, c being rational |

Equal in magnitude and opposite in sign | b = 0 |

Reciprocal to each other | c = a |

Both positive | b has a sign opposite to that of a and c |

Both negative | a, b, c all have same sign |

Opposite sign | a, c are of opposite sign |

→ Two equations a_{1}x^{2} + b_{1}x + c_{1} = 0, a_{2}x^{2} + b_{2}x + c_{2} = 0 have exactly the same roots if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

→ The equations a_{1}x^{2} + b_{1}x + c_{1} = 0, a2x^{2} + b_{2}x + c_{2} = 0 have a common root, if (c_{1}a_{2} – c_{2}a_{1})^{2} = (a_{1}b_{2} – a_{2}b_{1})(b_{1}c_{2} – b_{2}c_{1}) and the common root is \(\frac{c_{1} a_{2}-c_{2} a_{1}}{a_{1} b_{2}-a_{2} b_{1}}\) if a_{1}b_{2} ≠ a_{2}b_{1}

→ If f(x) = 0 is a quadratic equation, then the equation whose roots are

- The reciprocals of the roots of f(x) = 0 is f\(\left(\frac{1}{x}\right)\) = 0
- The roots of f(x) = 0, each ‘increased’ by k is f(x – k) = 0
- The roots of f(x) = 0, each ‘diminished’ by k is f(x + k) = 0
- The roots of f(x) = 0 with sign changed is f(-x) = 0
- The roots of f(x) = 0 each multiplied by k(≠0) is f\(\left(\frac{x}{k}\right)\) = 0

→ Sign of the expression ax^{2} + bx + c = 0

- The sign of the expression ax
^{2}+ bx + c is same as that of ‘a’ for all values of x if b^{2}– 4ac ≤ 0 i.e. if the roots of ax^{2}+ bx + c = 0 are imaginary or equal. - If the roots of the equation ax
^{2}+ bx + c = 0 are real and different i.e b^{2}– 4ac > 0, the sign of the expression is same as that of ‘a’ if x does not lie between the two roots of the equation and opposite to that of ‘a’ if x lies between the roots of the equation.

→ The expression ax^{2} + bx + c is positive for all real values of x if b^{2} – 4ac < 0 and a > 0.

→ The expression ax^{2} + bx + c has a maximum value when ‘a’ is negative and x = –\(\frac{\mathrm{b}}{2 \mathrm{a}}\). Maximum value of the expression = \(\frac{4 a c-b^{2}}{4 a}\)

→ The expression ax^{2} + bx + c has a maximum value when ‘a’ is positive and x = –\(\frac{\mathrm{b}}{2 \mathrm{a}}\). Minimum value of the expression = \(\frac{4 a c-b^{2}}{4 a}\)

Theorem 1:

If the roots of ax^{2} + bx + c = 0 are imaginary, then for x ∈ R , ax^{2} + bx + c and a have the same sign.

Proof:

The root are imaginary

b^{2} – 4ac < 0 4ac – b^{2} > 0

\(\frac{a x^{2}+b x+c}{a}=x^{2}+\frac{b}{a} x+\frac{c}{a}=\left(x+\frac{b}{2 a}\right)^{2}+\frac{c}{a}-\frac{b^{2}}{4 a^{2}}=\left(x+\frac{b}{2 a}\right)^{2}+\frac{4 a c-b^{2}}{4 a^{2}}\)

∴ For x ∈ R, ax^{2} + bx + c = 0 and a have the same sign.

Theorem 2.

If the roots of ax^{2} + bx + c = 0 are real and equal to α = \(\frac{-b}{2 a}\), then α ≠ x ∈ R ax^{2} + bx + c and a will have same sign.

Proof:

The roots of ax^{2} + bx + c = 0 are real and equal

⇒ b^{2} = 4ac ⇒ 4ac – b^{2} = 0

\(\frac{a x^{2}+b x+c}{a}\) = x + \(\frac{b}{a}\)x + \(\frac{c}{a}\)

= \(\left(x+\frac{b}{2 a}\right)^{2}+\frac{c}{a}-\frac{b^{2}}{4 a^{2}}\)

= \(\left(x+\frac{b}{2 a}\right)^{2}+\frac{4 a c-b^{2}}{4 a^{2}}\)

= \(\left(x+\frac{b}{2 a}\right)^{2}\) > 0 for x ≠ \(\frac{-b}{2 a}\) = α

For α ≠ x ∈ R, ax^{2} + bx + c and a have the same sign.

Theorem 3.

Let be the real roots of ax^{2} + bx + c = 0 and α < β. Then

(i) x ∈ R, α < x< β ax^{2} + bx + c and a have the opposite signs

(ii) x ∈ R, x < α or x> β ax^{2} + bx + c and a have the same sign.

Proof:

α, β are the roots of ax^{2} + bx + c = 0

Therefore, ax^{2} + bx + c = a(x – α)(x – β)

\(\frac{a x^{2}+b x+c}{a}\) = (x – α)(x – β)

(i) Suppose x ∈ R, α < x < β

⇒ x < α < β then x – α < 0, x – β < 0 ⇒ (x – α)(x – β) > 0 ⇒ \(\frac{a x^{2}+b x+c}{a}\) > 0

⇒ ax^{2} + bx + c, a have a same sign

(ii) Suppose x ∈ R, x > β, x > β > α then x – α > 0, x – β > 0

⇒ (x – α)(x – β) > 0 ⇒ \(\frac{a x^{2}+b x+c}{a}\) > 0

⇒ ax^{2} + bx + c, a have same sign

∴ x ∈ R, x < α or x > β ⇒ ax^{2} + bx + c and a have the same sign.