# Sum And Product Of Roots Of Quadratic Equations

A quadratic equation takes the general form ax^{2 }+ 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:

x^{2} – (sum of roots)x + (product of roots) = 0

Mathematically as; x^{2 }– (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 ax^{2 }+ bx + c = 0, we can proceed to make the coefficient of x^{2} unity by dividing through with the coefficient of x^{2}

ax^{2}/a + bx/a + c/a = 0

x^{2 }+ 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

- x
^{2 }– 8x – 4 = 0 - 4x
^{2 }+ 2x + 16 = 0

- ½ x
^{2 }– 10x -1 = 0

- 9x
^{2 }= 3x + 5 - x
^{2}+ 12x = 6

**Solutions**

If α and β are the roots of a quadratic equation ax^{2 }+ bx + c = 0, then α + β = -b/a and αβ = c/a. We shall apply this rule in all the equations here.

- x
^{2 }– 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

**Also Recommended:**

**Quadratic Equation: How To Solve Using Factorization Method**

**Completing The Square Method Of Quadratic Equations**

**Quadratic Equation Using Quadratic Formula Method**

**Simultaneous Equation- Meaning, Methods, and Examples**

**Changing The Subject Of Formula**

- 4x
^{2 }+ 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 x
^{2 }– 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

- 9x
^{2 }= 3x + 5

Move all numbers to the left-hand side of the equation. You must be mindful of the signs.

9x^{2 }– 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**

- x
^{2}+ 12x = 6

Move all numbers to the left-hand side of the equation.

x^{2} + 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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