Question

If α and β are the zeroes of the polynomial f(x) = 4x^2

– 6x + 2, then the value of α^2

– β^

2

is​

Answer 1

$\large\bf\underline{Given:-}$

• p(x) = 4x² - 6x + 2

$\large\bf\underline {To \: find:-}$

We need to find the value of α² - β²

$\huge\bf\underline{Solution:-}$

• ➤ p(x) = 4x² - 6x + 2
• a = 4
• b = -6
• c = 2

➛ Sum of zeroes = -b/a

➛ α + β = -(-6)/4

➛ α + β = 6/4

➛ α + β = 3/2

➛ Product of zeroes = c/a

➛ αβ = 2/4

➛ αβ = 1/2

we know that,

➧ (a - b)² = a² + b² - 2ab

➧ a -b = √a² + b² - 2ab ....(1)

and,

➧ (a + b)² = a² + b² + 2ab

➧ a² + b² = (a + b)² - 2ab

➧ a² + b² = (3/2)² - 2 × 1/2

➧ α² + β² = 9/4 -1

➧ α² + β² = (-4+9)/4

➧ α² + β² = 5/4.......(2)

and,

➧ (a + b)(a - b) = a² - b².....(3)

Now,

Finding value of α² - β²

➙ α² - β² = (α + β)(α - β)

From 1) , 2) and 3)

➙ α² - β² = (α + β)(√α² + β² - 2αβ)

➙ α² - β² = (3/2)(√5/4 - 2 × ½

➙ α² - β² = (3/2)(√5/4 -1)

➙ α² - β² = (3/2)(√(5 -4)/4)

➙ α² - β² = (3/2)(√1/4)

➙ α² - β² = 3/2 × 1/2

➙ α² - β² = 3/4

Hence,

▶️ Value of α² - β² = 3/4

$\rule{200}3$