If a quadratic equation is given in standard form, we can find the sum and product of the roots using coefficient of x2, x and constant term.
Let us consider the standard form of a quadratic equation,
ax2 + bx + c = 0
(Here a, b and c are real and rational numbers)
Let α and β be the two zeros of the above quadratic equation.
Then the formula to get sum and product of the roots of a quadratic equation is,
Example 1 :
Find the sum and product of roots of the quadratic equation given below.
x2 - 5x + 6 = 0
Solution :
Comparing
x2 - 5x + 6 = 0
and
ax2 + bx + c = 0
we get
a = 1, b = -5 and c = 6
Therefore,
Sum of the roots = -b/a = -(-5)/1 = 5
Product of the roots = c/a = 6/1 = 6
Example 2 :
Find the sum and product of roots of the quadratic equation given below.
x2 - 6 = 0
Solution :
Comparing
x2 - 6 = 0
and
ax2 + bx + c = 0
we get
a = 1, b = 0 and c = -6
Therefore,
Sum of the roots = -b/a = 0/1 = 0
Product of the roots = c/a = -6/1 = -6
Example 3 :
Find the sum and product of roots of the quadratic equation given below.
3x2 + x + 1 = 0
Solution :
Comparing
3x2 + x + 1 = 0
and
ax2 + bx + c = 0
we get
a = 3, b = 1 and c = 1
Therefore,
Sum of the roots = -b/a = -1/3
Product of the roots = c/a = 1/3
Example 4 :
Find the sum and product of roots of the quadratic equation given below.
3x2 + 7x = 2x - 5
Solution :
First write the given quadratic equation in standard form.
3x2 +7x = 2x - 5
3x2 + 5x + 5 = 0
Comparing
3x2 + 5x + 5 = 0
and
ax2 + bx + c = 0
we get
a = 3, b = 5 and c = 5
Therefore,
Sum of the roots = -b/a = -5/3
Product of the roots = c/a = 5/3
Example 5 :
Find the sum and product of roots of the quadratic equation given below.
3x2 -7x + 6 = 6
Solution :
First write the given quadratic equation in standard form.
3x2 -7x + 6 = 6
3x2 - 7x = 0
Comparing
3x2 - 7x = 0
and
ax2 + bx + c = 0
we get
a = 3, b = -7 and c = 0
Therefore,
Sum of the roots = -b/a = -(-7)/3 = 7/3
Product of the roots = c/a = 0/3 = 0
Example 6 :
Find the sum and product of roots of the quadratic equation given below.
x2 + 5x + 1 = 3x2 + 6
Solution :
First write the given quadratic equation in standard form.
x2 + 5x + 1 = 3x² + 6
0 = 2x2 - 5x + 5
2x2 - 5x + 5 = 0
Comparing
2x2 - 5x + 5 = 0
and
ax2 + bx + c = 0
we get
a = 2, b = -5 and c = 5
Therefore,
Sum of the roots = -b/a = -(-5)/3 = 5/2
Product of the roots = c/a = 5/2
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