FINDING UNKNOWN IN A QUADRATIC EQUATION WHEN ROOTS ARE GIVEN

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

Problem 1 :

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

Solution :

x² – kx + k – 1 = 0

a = 1, b = -k, c = k - 1

α = α, β = 2α

Applying (1) in (2), we get

2(k/3)² = k – 1

2k²/9 = k – 1

2k² = 9(k – 1)

2k² = 9k – 9

2k² – 9k + 9 = 0

(k – 3) (2k – 3) = 0

k = 3, 2k = 3

 k = 3/2

 So, the value(s) of k is 3 and 3/2.

Problem 2 :

The roots of the quadratic equation

x² + 6x + c are k and k - 1

Find the value of c.

Solution :

x² + 6x + c

a = 1, b = 6, c = c

α = k, β = k - 1

Applying (1) in (2), we get

(-5/2)² – (-5/2) = c

25/4 + 5/2 = c

25/4 + 5/2 × (2/2) = c

25/4 +10/4 = c

35/4 = c

So, the value of c is 35/4.

Problem 3 :

The roots of the quadratic equation

2x² - 9x + k

are m/2 and m - 3. Find the value of k.

Solution :

2x² - 9x + k

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

α = m/2, β = m - 3

So, the value of k is 10.

Problem 4 :

Find the values of m for which one root of the equation

4x² + 5 = mx

is three times the other root.

Solution :

4x² + 5 = mx

4x² - mx + 5 = 0

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

Let, the roots are α and 3α.

α = α, β = 3α

By applying (1) in (2), we get

(m/16)² = (5/12)

m²/256 = (5/12)

m² = 5/12 × 256

m² = 320/3

m =√(320/3)

So, the value of m is 8√(5/3).

Problem 5 :

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

Solution :

3x² - 4x + m = 0

a = 3, b = -4, c = 1

α = 2β, β = β

By applying (1) in (2), we get

(4/9)² = m/6

16/81 = m/6

16/81 × 6 = m

96/81 = m

32/27 = m

So, the value of m is 32/27.

Problem 6 :

The roots of the equation

4x² - kx + 35 = 0

differ by one. Find the value of k.

Solution :

4x² - kx + 35 = 0

a = 4, b = -k, c = 35

√((α + β)² - 4αβ) = 1

√((k/4)² – 4(35/4) = 1

√((k²/16) - 35) = 1

Taking square on both sides.

((k²/16) - 35) = 1

k²/16 = 1 + 35

k²/16 = 36

k² = 36 × 16

k² = 576

k = √576

k = 24

So, the value of k is 24.