**Category : **10th Class

If are the roots of the quadratic equation, then the relation between the roots of the quadratic equation is given by,

Sum of the roots

Product of the roots

** Formation of Quadratic Equations**

If are the roots of the quadratic equation, S denotes its sum and P denotes its product, then the quadratic equation is given by:

** Graphical Representation of a Quadratic Equation**

For the quadratic equation , the nature of graph for different values of D is: (a) If D < 0, and a > 0, then the graph is given by:

If a < 0, then the graph is given by.

(b) If D = 0, and a > 0, then the graph of the function is given by

If a < 0, then the graph is given by

(d) If D > 0, and a > 0, then the graph of the function is given by,

If a < 0, then the graph of the function is given by,

** Roots of Biquadratic Equation**

Any biquadratic equation, , will have four roots. If a, P, Y, and 5 are its roots, then the relation between the roots is given by Sum of the roots

Sum of product of two roots at a time

Sum of product of three roots at a time

Product of the roots

**Find the value of.**

(a)

(b)

(c)

(d)

(e) None of these

**Answer:** (a)

**Explanation **

Let

**A If are the roots of the equation, then the value of are.**

(a)

(b)

(c)

(d)

(e) None of these

**Answer:** (b)

**Explanation**

We have,

** If is real and is the roots of , then find the value of .**

(a)

(b)

(c)

(d)

(e) None of these

**Answer:** (d)

**Explanation **

We have sum of the roots

Product of the roots

Form the above two equations we get,

** If are the roots of and are the roots of, then find the value of .**

(a) 15

(b) 33

(c) 41

(d) 63

(e) None of these

**Answer:** (b)

**If the coefficient of z in the quadratic equation is taken as 18 in place of 12 and its roots were found to be -16 and -2. The roots of the original equation are:**

(a) - 12 & - 6

(b) - 14 & - 4

(c) - 16 & - 2

(d) ? 8 &- 4

(e) None of these

**Answer:** (d)

- The demerit of the factorization method, is that it takes time to figure out the numbers within the factors.
- The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt.
- The Hindu mathematician Aryabhata (475 or 476-550) gave a rule for the sum of a geometric series that shows knowledge of the quadratic equations with both solutions
- Viete was among the first to replace geometric methods of solution with analytic ones, although he apparently did not grasp the idea of a general quadratic equation.
- Sridhara gave the positive root of the quadratic formula, as stated by Bhaskara.

- The most general quadratic equation is in the form, where a, b, c are constants and x is the variable.
- A real number m is said to be the root of the quadratic equation if it satisfies the quadratic equation, .
- For any quadratic equation, the number of root is always two.
- Quadratic equation can be solved by factorization or using the discriminant method.
- The value of .
- No real root exist for D < 0.

*play_arrow*Introduction*play_arrow*Quadratic Equations*play_arrow*Relations between Roots of Quadratic Equation

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