Question

3. If a and B are zeroes of the polynomial f(x)=4x--5x-1, then
find alpha^2+beta^2

Answer 1

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

• α and β are zeroes of f(x) = 4x² - 5x - 1

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

• Value of α² + β²

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

• f(x) = 4x² - 5x - 1

a = 4

b = -5

c = -1

• sum of zeroes = -b/a

≫ α + β = 5/4

• Product of zeroes = c/a

≫ αβ = -1/4

we know that,

• (a+b)² = a² + b² + 2ab
• a² + b² = (a+ b)² - 2ab

Now,

finding value of α² + β²

»» α² + β² = (5/4)² -2×(-1/4)

»» α² + β² = 25/16 + 1/2

»» α² + β² = (25+8)/16

»» α² + β² = 33/16

So,

Value of α² + β² = 33/16.

Answer 2

$\Large{\underline{\underline{\mathfrak{\bf{\red{Solution}}}}}}$

$\Large{\underline{\mathfrak{\bf{\orange{Given}}}}}$

• α & β are zeroes of equation, f(x) = 4x² - 5x - 1 .

$\Large{\underline{\mathfrak{\bf{\orange{Find}}}}}$

• Value of ( α² + β²).

$\Large{\underline{\underline{\mathfrak{\bf{\red{Explanation}}}}}}$

$\large{\underline{\mathfrak{\bf{\orange{Using\:Formula}}}}}$

$\small\boxed{\tt{\green{\:Sum\:of\:Zeroes\:=\:\frac{\:Coefficient\:of\:x}{\:Coefficient\:ofx^2}}}}$

$\small\boxed{\tt{\green{\:product\:of\:zeroes\:=\:\frac{\:constant\:part}{\:Coefficient\:of\:x^2}}}}$

Then,

➩ Sum of zeroes = -(-5)/4

➩ α + β = 5/4 -------------(1)

again,

➩Product of zeroes = -1/4

➩ α . β = -1/4 ---------------(2)

Now, Squaring both Side of Equation(1)

➩ (α + β)² = (5/4)²

➩ (α² + β² + 2α β ) = 25/16

Keep Value by equ(2)

➩ (α² + β²) = 25/16 - 2 × (-1/4)

➩ (α² + β²) = 25/16 + 1/2

➩ (α² + β²) = (25+8)/16

➩ (α² + β²) = 33/16

______________________

$\Large{\underline{\underline{\mathfrak{\bf{\red{Hence}}}}}}$

• Value of (α² + β²) be = 33/16

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