Sum And Product Of Roots Of Quadratic Equations
A quadratic equation takes the general form ax2 + bx + c = 0. Given that the roots of the quadratic equation are α and β, then the sum and product of the roots are:
Sum of roots = (α + β)
Product of roots = αβ
Therefore, the general formula for a quadratic equation can be expressed as:
x2 – (sum of roots)x + (product of roots) = 0
Mathematically as; x2 – (α + β)x + αβ = 0
Sometimes, you may be required to find the sum and product of the roots of a quadratic equation without solving the equation right away. This article will give you a detailed step-by-step guide to finding the sum and product of roots of quadratic equations easily. Ensure that you follow all the steps accordingly.
How To Find The Sum and Product Of Roots Of Quadratic Equations
You must know the formula to find the sum and product of roots. We can derive the formula as follows:
Given that α and β are the roots of a quadratic equation ax2 + bx + c = 0, we can proceed to make the coefficient of x2 unity by dividing through with the coefficient of x2
ax2/a + bx/a + c/a = 0
x2 + bx/a + c/a = 0
Sum of roots (α + β) = – coefficient of x
= -b/a
Product of root (αβ) = constant term
= c/a
We shall further illustrate this using some examples that have been worked on.
Worked Examples
Find the sum and product of roots of the following quadratic equations
- x2 – 8x – 4 = 0
- 4x2 + 2x + 16 = 0
- ½ x2 – 10x -1 = 0
- 9x2 = 3x + 5
- x2 + 12x = 6
Solutions
If α and β are the roots of a quadratic equation ax2 + bx + c = 0, then α + β = -b/a and αβ = c/a. We shall apply this rule in all the equations here.
- x2 – 8x – 4 = 0
Comparing it with the general quadratic equation, a = 1, b = -8 and c = -4
α + β = -b/a = -(-8)/1
= 8
α β = c/a = -4/1
= -4
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Changing The Subject Of Formula
- 4x2 + 2x + 16 = 0
Comparing it with the general quadratic equation, a = 4, b = 2 and c = 16
α + β = -b/a = 2/4
= ½
α β = c/a = 16/4
= 4
- ½ x2 – 10x -1 = 0
Comparing it with the general quadratic equation, a = ½, b = -10 and c = -1
α + β = -b/a = -(-10)/½
= 20
α β = c/a = -1/½
= -2
- 9x2 = 3x + 5
Move all numbers to the left-hand side of the equation. You must be mindful of the signs.
9x2 – 3x – 3 = 0
Comparing it with the general quadratic equation, a = 9, b = -3 and c = -3
α + β = -b/a = -(-3)/9
= 1/3
α β = c/a = -3/9
= -1/3
Further Solutions
- x2 + 12x = 6
Move all numbers to the left-hand side of the equation.
x2 + 12x – 6 = 0
Comparing it with the general quadratic equation, a = 1, b = 12 and c = -6
α + β = -b/a = -12/1
= -12
α β = c/a = -6/1
= –6
Finding the sum and product of the roots of a quadratic equation is easy and straightforward when you know the proper steps. You can do more exercises on this.
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