Back to: Further Mathematics SS2
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 =
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
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.
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