 FINDING QUADRATIC EQUATION GIVEN SUM AND PRODUCT OF ROOTS CONDITION FOR EQUAL ROOTS, REAL ROOTS, AND NO ROOT

Hence the roots of the equation are 6 and -7. In general, if a quadratic equation factorizes into

(x – α) (x – β) = 0

then α and β must be the roots of that equation.

The general quadratic equation ax2 + bx + c = 0 can also be written as:

x2 +                                                …(1)

If the roots of the equation are α and β then the above equation can be written as:

(x –α) (x – β) = 0

x2 – (α – β) x + αβ = 0                                    —(2

By comparing coefficients in equations (1) and (2)

: α + β =

andαβ =

The above consideration gives rise to two problems:

(a) Given a quadratic equation, we can find the sum and product of the roots.

(b) Given the roots, we can formulate the corresponding quadratic equation.

The quadratic equation whose roots are α and β is

x2 – (α + β) x + α β = 0

Find the sum and product of the roots of each of the following quadratic equations:

(a) 2x2 + 3x – 1 = 0

(b) 3x2 – 5x – 2 = 0

(c) x2 – 4x – 3 = 0

(d) ½ x2 – 3x – 1 = 0

Find the quadratic equation whose roots are:

(a) 3 and -2                                         (b) ½ and 5

(c) -1 and 8                                          (d)¾ and ½

Solution

The quadratic equation whose roots are α and β is x2 – (α + β) x +α β = 0.

(a) α + β = 3 – 2 = 1, α β = 3 (-2) = -6

: The quadratic equation whose roots are 3 and -2 is x2 – x – 6 = 0.

Symmetric Properties of Roots

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