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# Quadratic Equation, Algebra, Math Assignment Help

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mathematics - Quadratic Equation, Algebra, Math

**General Quadratic Equation:** An equation of the form

ax^{2} + bx + c = 0

where a ¹ 0, is called a quadratic equation, in the real or complex coefficients a, b and c.

** **

**Roots of a Quadratic Equation: **

The values of x (say x = a, ß) which satisfy the quadratic equation (1) are called the roots of the equation and they are given by

a = -b+ (a^{2} - 4ac) / 2a

ß = -b- (a^{2} - 4ac) / 2a

**Discriminant of a Quadratic Equation:** The quantity D = b^{2} - 4ac, is known as the discriminant of the equation.

**Nature of the Roots:** In quadratic equation ax^{2} + bx + c = 0, let us suppose that a, b, c are real and a ¹ 0. The following is true about the nature of its roots then

(i) The equation has real and distinct roots if and only if D = b^{2} - 4ac > 0.

(ii) The equation has real and coincident (equal roots if and only if D = b^{2} - 4ac = 0.

(iii) The equation has rational roots if and only if a, b, CEQ (the set of rational numbers) and D = b^{2} - 4ac is a perfect square (of a rational number).

(iv) The equation has (unequal) irrational (surd form) roots if and only if D = b^{2} - 4ac > 0 and not a perfect square even if a, band c are rational. In this case if p + q^{1/2}, p, q rational is an irrational root, then p-q^{1/2} is also a root (a, b, c being rational).

(v) a + ib (b ¹ 0 and a, b Î R) is a root if and only if its conjugate a - ib is a root, that is complex roots always occur in conjugate pairs in a quadratic equation. In case the equation is satisfied by more than two complex numbers, then it reduces to an identity.

0. x^{2} + 0. x + 0 = 0, i.e., a = 0 = b = c.

**Relation between Roots and Coefficients:** If a, ß are the roots of the quadratic equation ax^{2} + bx + c = 0, then the sum and product of the roots is

a**+** ß** = -**b/a and a ß = C/A

Hence the quadratic equation whose roots are a and ß is given by

x^{2} - (a + ß) x + aß = 0 or (x - a) (x - ß) = 0.

**Condition that the Two Quadratic Equations have a Common Root: **

Thus eliminating a., the condition for a common root is given by

(c_{1}a_{2} - c_{2}a_{1})^{2} = (b_{1}c_{2} - b_{2}c_{1}) (a_{1}b_{2}- a_{2}b_{1})

It is to be noted here that two different quadratic equations with rational coefficients cannot have a common root which is non-real complex or irrational, as imaginary and surd roots always occur in pairs.

**Condition that the Two Quadratic Equations have both the Roots Common:** The two quadratic equations will have the same roots if and only if their coefficients are proportional, i.e.,

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

Condition that one root of a quadratic equation may be the square of the other root, i.e. the roots are a and ß = a^{2} is b^{3} + ca^{2} + ac^{2} = 3abc.

**Higher Degree Equation:** The equation

p(x) = a_{0}x^{n} + a_{1}x^{n-1} + .......... + z_{n-1} x + a_{n} = 0

Where the coefficients a_{0}, a_{1}, ........, an Î R (or C) and a_{0} ¹ 0 is called an equation of nth degree, which has exactly n roots a_{1}, a_{2}, ... , a_{n} Î C.

Σa_{1} = a_{1}, + a_{2} + ........ + a_{n} = -a_{1} / a_{0}

Σa_{1} a_{2} = a_{1}a_{2} + ........... + a_{n-1}, a_{n} = a_{0}/a_{1}

and so on and a_{1},a_{2} ......... a_{n} = (-1)^{n} a_{n}/a_{0}

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**General Quadratic Equation:**An equation of the form

^{2}+ bx + c = 0

**Roots of a Quadratic Equation:**

^{2}- 4ac) / 2a

^{2}- 4ac) / 2a

**Discriminant of a Quadratic Equation:**The quantity D = b

^{2}- 4ac, is known as the discriminant of the equation.

**Nature of the Roots:**In quadratic equation ax

^{2}+ bx + c = 0, let us suppose that a, b, c are real and a ¹ 0. The following is true about the nature of its roots then

^{2}- 4ac > 0.

^{2}- 4ac = 0.

^{2}- 4ac is a perfect square (of a rational number).

^{2}- 4ac > 0 and not a perfect square even if a, band c are rational. In this case if p + q

^{1/2}, p, q rational is an irrational root, then p-q

^{1/2}is also a root (a, b, c being rational).

^{2}+ 0. x + 0 = 0, i.e., a = 0 = b = c.

**Relation between Roots and Coefficients:**If a, ß are the roots of the quadratic equation ax

^{2}+ bx + c = 0, then the sum and product of the roots is

**+**ß

**= -**b/a and a ß = C/A

^{2}- (a + ß) x + aß = 0 or (x - a) (x - ß) = 0.

**Condition that the Two Quadratic Equations have a Common Root:**

_{1}a

_{2}- c

_{2}a

_{1})

^{2}= (b

_{1}c

_{2}- b

_{2}c

_{1}) (a

_{1}b

_{2}- a

_{2}b

_{1})

**Condition that the Two Quadratic Equations have both the Roots Common:**The two quadratic equations will have the same roots if and only if their coefficients are proportional, i.e.,

_{1}/a

_{2}= b

_{1}/b

_{2}= c

_{1}/c

_{2}

^{2}is b

^{3}+ ca

^{2}+ ac

^{2}= 3abc.

**Higher Degree Equation:**The equation

_{0}x

^{n}+ a

_{1}x

^{n-1}+ .......... + z

_{n-1}x + a

_{n}= 0

_{0}, a

_{1}, ........, an Î R (or C) and a

_{0}¹ 0 is called an equation of nth degree, which has exactly n roots a

_{1}, a

_{2}, ... , a

_{n}Î C.

_{1}= a

_{1}, + a

_{2}+ ........ + a

_{n}= -a

_{1}/ a

_{0}

_{1}a

_{2}= a

_{1}a

_{2}+ ........... + a

_{n-1}, a

_{n}= a

_{0}/a

_{1}

_{1},a

_{2}......... a

_{n}= (-1)

^{n}a

_{n}/a

_{0}