FIND QUADRATIC POLYNOMIAL WITH GIVEN SUM AND PRODUCT OF ZEROES

Find quadratic polynomial for each of the following pairs of values where the first value stands for sum of zeroes and the second value stands for product of zeroes.

1) 3, 1

2) -2, 4

3) 0, 4

4) √2, 1/5

5) -1/3, 1

6) 1/2, -4

7) 1/3, -1/3

8) -√3, 2 

Use the sum and product of roots formulas to answer the questions below:

9)  The roots of the equation x² - kx + k - 1 = 0 are α and 2α. Find the value(s) of k.

10)  The roots of the quadratic equation x² + 6x + c are k and k – 1. Find the value of c.

11)  The roots of the quadratic equation  2x² - 9x + k are m/2 and m - 3. Find the value of k.

12)  Find the values of m for which one root of the equation 4x² + 5 = mx is three times the other root.

13)  One root of the equation 3x² - 4x + m = 0 is double the other. Find the roots and the value of m

1. Answer :

3, 1

sum of zeroes = 3, product of zeroes = 1

Quadratic polynomial :

x² - 3x + 1

2. Answer :

-2, 4

sum of zeroes = -2, product of zeroes = 4

Quadratic polynomial :

x² + 2x + 4

3. Answer :

0, 4

sum of zeroes = 0, product of zeroes = 4

Quadratic polynomial :

x² + 4

4. Answer :

 √2, 1/5

sum of zeroes =  √2, product of zeroes = 1/5

Quadratic polynomial :

x² - √2x + (1/5)

5. Answer :

-1/3, 1

sum of zeroes = -1/3, product of zeroes = 1

Quadratic polynomial :

x² + x/3 + 1

6. Answer :

1/2, -4

sum of zeroes = 1/2, product of zeroes = -4

Quadratic polynomial :

x² - x/2 - 4

7. Answer :

1/3, -1/3

sum of zeroes = 1/3, product of zeroes = -1/3

Quadratic polynomial :

x² - x/3 - 1/3

8 Answer :

-√3, 2

sum of zeroes = -√3, product of zeroes = 2

Quadratic polynomial :

x² + √3x + 2

9 Answer :

The given equation is x² - kx + k - 1 = 0

The roots are α and 2α

Here a = 1, b = -k and c = (k - 1)

Sum of roots = -b/a

α + 2α = -k/1

3α = -k/1

k = -3α

Product of roots = c/a

α (2α) = (k - 1)/1

2α² = k - 1

Applying the value of k, we get

2α² = 3α - 1

2α² - 3α + 1 = 0

2α² - 2α - 1α + 1 = 0

2α(α - 1) - 1(α - 1) = 0

(2α - 1)(α - 1) = 0

2α - 1 = 0 and α - 1 = 0

α = 1/2 and α = 1

So, the values of k are -3/2 and -3.

10. Answer :

Given quadratic equation is x² + 6x + c

Roots are k and k – 1

a = 1, b = 6 and c = c

Sum of roots = -b/a

k + k - 1 = -6/1

2k - 1 = -6

2k = -6 + 1

2k = -5

k = -5/2

Product of roots = c/a

k(k - 1) = c/1

-5/2(-5/2 - 1) = c

c = -5/2 (-7/2)

= 35/4

So, the value of c is 35/4.

11. Answer :

Given equation that,  2x² - 9x + k

the roots of the quadratic equation are m/2 and m - 3

a = 2, b = -9 and c = k

Sum of roots = -b/a

(m/2) + (m - 3) = -(-9)/2

(m/2) + (m - 3) = 9/2

[m + 2(m - 3)]/2 = 9/2

m + 2m - 6 = 9

3m - 6 = 9

3m = 12

m = 12/3

m = 4

Product of roots = c/a

(m/2) (m - 3) = k/2

applying the value of m, we get

(4/2)(4 - 3) = k/2

2(1) = k/2

k = 2(2)

k = 4

12. Answer :

4x² + 5 - mx = 0

4x² - mx + 5 = 0

Let α be one root, then the other root will be 3α.

a = 4, b = -m and c = 5

Sum of roots = -b/a

α + 3α = -(-m)/4

4α = m/4

α = m/16

Product of roots = c/a

α (3α) = 5/4

(m/16) (3m/16) = 5/4

3m²/256 = 5/4

3m² = (5/4)(256)

3m² = (5)(64)

m² = (5)(64)/3

m = √(5)(64)/3

m = ±8√(5/3)

13. Answer :

3x² - 4x + m = 0

Let α be one root, then the other root will be 2α.

a = 3, b = -4 and c = m

Sum of roots = -b/a

α + 2α = 4/3

3α = 4/3

α = 4/9

Product of roots = c/a

α (2α) = m/3

2α² = m/3

Applying the value of α, we get

2(4/9)² = m/3

2(16/81) = m/3

m/3 = 32/81

m = 32/27

So, the value of m is 32/27.

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