Grade 11Algebra

Image attached ..... .. .. ... .. .. ..... ... .. ...........................

mahima pareek

7 Years agoGrade 11

Samyak Jain

7 Years ago

Let us asume $\alpha$ and $\beta$ are distinct roots of x² + px + q = 0.

$\therefore$ $\alpha$ + $\beta$ = – p ...(1) & $\alpha$ $\beta$ = q ...(2)

$\because$ $\alpha$ and $\beta$ are real and distinct roots, discriminant D > 0.

$\Rightarrow$ p² – 4q > 0 …..(3)

$\alpha$ ⁴ and $\beta$ ⁴ are roots of x² – rx + s = 0.

$\therefore$ $\alpha$ ⁴ + $\beta$ ⁴ = r & $\alpha$ ⁴ $\beta$ ⁴ = s

$\alpha$ ⁴ + $\beta$ ⁴ = ( $\alpha$ ² + $\beta$ ²)² – 2 $\alpha$ ² $\beta$ ² = [( $\alpha$ + $\beta$ )² – 2 $\alpha$ $\beta$ ]² – 2( $\alpha$ $\beta$ )²

= (p² – 2q)² – 2q² = p⁴ – 4p²q + 4q² – 2q²

= p⁴ – 4p²q + 2q² = p²(p² – 4q) + 2q² [From (1) & (2)]

$\alpha$ ⁴ $\beta$ ⁴ = ( $\alpha$ $\beta$ )² = q²

$\therefore$ r = p²(p² – 4q) + 2q² & s = q²

$\alpha$ ⁴ + $\beta$ ⁴ > 0 ; $\alpha$ ⁴ $\beta$ ⁴ > 0 for all $\alpha$ , $\beta$ i.e. r = p²(p² – 4q) + 2q² > 0 ; s>0

Now, consider x⁴ – 4qx + 2q² – r = 0

D = (-4q)² – 4(2q² – r) = 4(4q² – 2q² + r) = 4(2q² + r) > 0 $\because$ q²> 0 & r >0

This shows that roots of x⁴ – 4qx + 2q² – r = 0 are real.

Let f(x) = x⁴ – 4qx + 2q² – r

f(0) = 2q² – r = 2q² – [p²(p² – 4q) + 2q²] = 2q² – p²(p² – 4q) – 2q²

= – p²(p² – 4q)

From (3), p² – 4q > 0 $\Rightarrow$ p²(p² – 4q) > 0 $\Rightarrow$ – p²(p² – 4q)

i.e. f(0)

Graph of f(x) is concave upwards (because coefficient of x² is 1 > 0)

with the value of f(x) at x = 0 negative.

This is possible when one root is positive while other is negative.

Image attached ..... .. .. ... .. .. ..... ... .. ...........................

mahima pareek

1 Answer

Samyak Jain

What is a binomial expression?

Can we divide two vectors? If this is possible then how?

How do you write “the product of 18 and q” as an algebraic expression?

दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे

Find the least number of square tiles by which the floor of a room of dimensions 16.58 m and 8.32 m can be covered completely. why HCF and not LCM as “least” keywor dis given?

Image attached ..... .. .. ... .. .. ..... ... .. ...........................

mahima pareek

1 Answer

Samyak Jain

Other Related Questions on algebra

What is a binomial expression?

Can we divide two vectors? If this is possible then how?

How do you write “the product of 18 and q” as an algebraic expression?

दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे

Find the least number of square tiles by which the floor of a room of dimensions 16.58 m and 8.32 m can be covered completely. why HCF and not LCM as “least” keywor dis given?