Sum and Product of Roots

Explanation

In this equation, x² - 7x + 10 = 0
Here, a = 1, b = -7 and c = 10

1. Finding sum of roots:
   \(\text{Sum of Roots} = -\frac{b}{a} = - {-7 \over 1} = 7\)
    
2. Finding product of roots
   \(\text{Product of Roots} = \frac{c}{a} = {10 \over 1} = 10\)

Parent Tip: Ask your child to be careful with signs. Remember, multiplication of two negative numbers give positive values.

Explanation

​​​​​​Given Equation: x² - 6x + 9

Here, a = 1, b = -6 and c = 9

Using the standard formula 

• Sum of roots = \(-\frac{b}{c} = -\frac{(-6)}{9} = \frac{2}{3}\)
• Product of roots = \(\frac{c}{a} = \frac{9}{1} = 9\)

Explanation

Given Equation: 2x² -4x + 1 = 0

Here, a = 2, b = -4 and c = 1

Finding sum and products of roots

• \(\text{Sum of Roots} = -\frac{b}{a} = - {-4 \over 2} = 2\)
• \(\text{Product of Roots} = -\frac{c}{a} = {1 \over 2} = 0.5\)

Explanation

Given Equation: \( 12x² - 7x + 1 = 0\)

Here, a = 1, b = -7, and c = 12

• Sum of roots = 𝛂 + 𝛽 = \(- {b \over a} = - {(-7) \over 1} = 7\)
•  Product of Roots = 𝛂 × 𝛽 = \({c\over a} = {12\over 1} = 12\)

New roots are \({ 1 \over 𝛂} \ and  \ {1 \over 𝛽}\)

Finding the sum and product of the new roots

• \(Sum=  {1 \over 𝛂} +  {1 \over 𝛽} =  {{𝛽 + 𝛂} \over 𝛽𝛂} = {7 \over 12}\)
• Product \(=  {1\over 𝛂} ×  {1 \over 𝛽} = {1 \over 𝛂𝛽} = {1 \over 12}\)

Using the standard formula, the equation becomes
x² - (sum)x + product = 0
\(x^2 - {7 \over 12} x + {1 \over 12} = 0\)

Multiply through by 12 to remove fractions:
\(12 \times \{x^2 - {7 \over 12} x + {1 \over 12}\} \\ = 12x^2 - 7 x + 1\)

Tip: Children can form the quadratic equation with the help of sum and product of the roots. The standard equation can be as \(x^2 + \text{(sum of roots)}x + \text{product of roots}\)

Explanation

Given Equation: 5x² + 5x + 1 = 0

• Finding the sum and product of roots
  𝛂 + 𝛽 = −5
  αβ = 5

• Finding the sum and product of new roots
  Sum \(=  {1 \over 𝛂} + { 1 \over 𝛽 } = {{𝛽 \ + \ 𝛂 } \over 𝛽𝛂 } = {-5 \over 5} = -1\)
  Product \(=  {1 \over 𝛂} ×  {1 \over 𝛽} = {1 \over 𝛂 𝛽} = {1 \over 5}\)

• Form the new equation
  \(x^2 -(sum)x + product = 0\\ x^2 - (-1)x + {1\over 5} = 0\\ x^2 + x + {1\over 5} = 0\)

Note: Terminate any fractional terms if present in the equation.

• Multiply the equation by 5 to eliminate the denominator
  \(5x^2 + 5x + 1 = 0\)

Parent Tip: Use real life items to explain the difference between fractions and whole number to your child.

Explanation

Given Equation: 4x² - 16x + 15 = 0

• Finding sum and products of roots

1. Sum: 𝛂 + 𝛽 = 32
2. Product: αβ =  52

• Now form the new roots 

1. Sum \(= { 3 \over 2} + { 5 \over 2 } = {8 \over 2} = 4\)
2. Product \(=  {3 \over 2} × { 5 \over 2} = {15 \over 4 }\)

• Form the new quadratic equation
  \(x^2 -(sum) x + product = 0\\  x ^2 -4x + {15 \over 4 } = 0\)

• Multiply the terms by 4 to eliminate the fraction
  \(4 \times \{  x ^2 -4x + {15 \over 4 } = 0 \} \\=  4x ^2 -16x + {15}\)

Explanation

Given Equation: x² -5x + 6 = 0

Let's suppose a and b are the two roots of the given equation

Now, the sum and product are:

• Sum of roots = a + b =  5
• Product of roots= ab = 6

Now, the roots of new equations is a² and b², then the sum and product can be given as

• Sum = a² + b² 
  \(a^2 + b^2 = (a + b)^2 - 2ab\)  [*]
  \(a^2 + b^2 = 5^2 - 2\times 6\\ = 25 - 12 = 13\)
   
• Product = a² × b² 
  \(a^2 \times b^2 = (ab)^2 \\ = (6)^2 = 36\)

The new equation is:

\(x^2 + 13 + 36\)

Note: (*) The property used to find \(a^2 + b^2 \ is\) \( (a + b)^2 = a^2 + b^2 + 2ab\).

Explanation

Given Equation: 6x² + x - 12 = 0

Here a = 6, b = 1 and c = -12

• Sum of roots = \(-{b \over a} = - {1\over 6} \)
• Product of roots = \({c \over a} = {-12\over 6}  = -2\)

Explanation

Given Equation:  2x² + 3x + 1 = 0

Here, a = 2, b = 3, and c = 1

• Sum = - 3/2 
• Product = 1/2  

Explanation

Given Equation: 5x² - 15x + 50 = 0

Here a = 5, b = 15, and c = 50

Calculating sum and product of roots

• Sum of roots = - b/a = - 15/5 = 3
• Product of roots = c/a =  50/5 = 10

Explanation

Given Equation:  x² + 12x + 36 = 0

Here, a = 1, b = 12, and c = 36

Using formula for sum and product.

• Sum of roots = - b/a = - 12/1 = -12
• Product of roots = c/a =  36/1 = 36

Explanation

Given Equation: x² -8x + 15 = 0

Here, a = 1, b = -8, and c = 15

1. Sum of roots = - b/a = - (-8/1) = 8
2. Product of roots = c/a =  15/1 = 15

Explanation

Given Equations: x² - x - 20 = 0

Here, a = 1, b = -1, and c = -20

1. Sum of roots = \(- b/a = - (-1/1) = 1\)
2. Product of roots = \(c/a =  -20/1 = -20\)

Explanation

Given Equation:  x² + px + 6 = 0

Here a = 1, b = p, and c = 6

1. Sum of roots = \({- b \over a} = {- p \over 1} = -p\).
   Given sum = 5, -p = 5, so p = -5
    
2. Product = \({ c \over a} =  {6\over 1} = 6\)

Explanation

Given Equation: 7x² - 2x - 14 = 0 

Here, a = 7, b = -2, and c = -14

1. Sum of roots = \({- b \over a} = -{ -2 \over 7} = {2 \over 7}\)
2. Product of roots \(= {c\over a} =  {-14 \over 7} = -2\)