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Introduction |
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An equation of the form ax2+bx+c=0 where a, b, c are real numbers and where "a" does not equal to zero(0). |
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Some Basic Definitions |
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Notation: Usually a polynomial is denoted by P(x) and if k is any number then P(k) denotes the value of P(x) at x=k. |
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The degree of a polynomial: If ao
? 0, then the polynomial P(x) or f(x) is
of degree 'n' i.e., it is the highest power of the variable x in the rational and integral polynomial. A polynomial of degree 2 is called a quadratic polynomial. |
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Polynomial equation: An equation in a single variable is called a polynomial equation of degree 'n' if it is an equation of the form.
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Algebraic identity: An algebraic identity is a statement of equality between two algebraic expressions, but it is satisfied for all values of the variable. |
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Solution of an equation: The determination of all the roots of a given equation is called the solution of the equation. |
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There are four methods of solving quadratic equations.
i) By factorization
ii) By completing the squares
iii) By using the formula
iv) By graphing |
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Methods of completing a square and to derive the formula for the solution of the quadratic equation |
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There are two methods to derive the formula for the solution of the quadratic equation (i) Simple Method and (ii) By Sridhar's Method.
The quadratic equation has two roots. |
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Nature of the roots |
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Without solving the quadratic equation, the nature of the roots can be determined using the discriminant.
i) D>0 i.e., positive and not a perfect square. The roots are real and distinct (irrational).
ii) D>0 i.e., perfect square. The roots are rational and distinct.
iii) D=0 i.e, the roots are equal and each is equal to -(b/(2xa)).
iv) D < 0 i.e., negative. The roots are complex and always occur in conjugate pairs. Where D = b2 - 4ac. |
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Relation between the roots of a quadratic equation |
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Our investigation reveals that there is a definite relationship between the roots of a quadratic equation and the coefficient of the second term and the constant term.
The sum of the roots of a quadratic equation is equal to the negation of the
coefficient of the second term divided by the leading coefficient.
The product of the roots of a quadratic equation is equal to the
constant term divided by the leading coefficient. |
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Equations reducible to quadratic form |
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Recall, a quadratic equation is of the form . An equation is said to be reducible to quadratic (or of quadratic form) if the variable factor of the leading term is the square of the variable factor in the second variable term.
We can solve these types of equations if we make an appropriate substitution to make them appear quadratic. This process is often referred to as u substitution.
To solve equations of quadratic form:
1. Make an appropriate substitution so that the equation has been reduced to a quadratic equation. (Make sure you note what substitution you have made.)
2. Solve the quadratic equation obtained in step 1.
3. Use the values obtained in step 2 to obtain the values of the original variable you were asked to solve for.
4. Check your answers in the original equation. Discard any solutions which do not make true equations.
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Symmetric Functions |
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Any expression f(a,b) involving two numbers a and b is said to be symmetric if it remains unchanged when a and b are interchanged.
[i.e. if f(a,b) = f(b,a)]. |
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Formation of quadratic equations from given roots and conditions |
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i) The quadratic equations whose roots are a and b is
where S = sum of roots and P = product of roots |
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ii) Quadratic equations with real coefficients, the complex roots always occur in conjugate pairs.
i.e., a + ib and a - ib |
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Summary |
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An expression of the form a0xn
+ a1xn-1+....+ an
= 0, where n is a positive integer and a0, a1,...,an belong to
some number system F, is called a polynomial in the variable x over F.
A quadratic equation has exactly two roots.
For the quadratic equations, we have
i) b2 - 4ac > 0 Roots are real and distinct
ii) b2 - 4ac = 0 Roots are real and equal
iii) b2 - 4ac < 0 Roots are imaginary and distinct.
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