2(a). If alpha and beta are the root of the equation 2x^2-7x+4=0 find the equation whose roots are a^2 and b^2
2(b). Without using mathematical table or calculator, evaluate (2ab+b)^2 - (b-2a)^2
2(a). To find the equation whose roots are a^2 and b^2, we can use the fact that if a and b are roots of an equation, then x = a and x = b satisfy that equation.
Since alpha and beta are the roots of 2x^2-7x+4=0, we have:
alpha + beta = 7/2 (sum of roots)
alpha * beta = 4/2 = 2 (product of roots)
Substituting x = a^2 and x = b^2, we can find the equation with roots a^2 and b^2:
(a^2) + (b^2) = 7/2
(a^2) * (b^2) = 2
Expanding the equation and simplifying, we get:
(a^2 + b^2)^2 - 2(a^2)(b^2) = (7/2)^2 - 2(2)
a^4 + 2(a^2)(b^2) + b^4 - 4 = 49/4 - 4/1
a^4 + 2(a^2)(b^2) + b^4 - 4 = 37/4
Therefore, the equation whose roots are a^2 and b^2 is:
x^2 + 2(a^2)(b^2) + (b^2)^2 - 4 = 37/4.
2(b). We can simplify the expression (2ab+b)^2 - (b-2a)^2 without using a mathematical table or calculator.
(2ab+b)^2 - (b-2a)^2 = (2ab+b)^2 - (b^2 - 4ab + 4a^2)
Expanding both terms, we get:
(4a^2b^2 + 4ab + b^2) - (b^2 - 4ab + 4a^2)
Simplifying, we cancel out like terms:
4a^2b^2 + 4ab + b^2 - b^2 + 4ab - 4a^2
Combining like terms, we have:
4a^2b^2 + 8ab - 4a^2
Therefore, without using a mathematical table or calculator, (2ab+b)^2 - (b-2a)^2 equals 4a^2b^2 + 8ab - 4a^2.
Let's solve the questions step by step:
2(a). To find the equation whose roots are α² and β², we can start by finding the sum and product of the roots of the quadratic equation 2x² - 7x + 4 = 0.
The sum of the roots (α + β) can be found using the formula:
α + β = -b/a
In this equation, a = 2 and b = -7, substituting these values:
α + β = -(-7)/2 = 7/2
The product of the roots (αβ) can be found using the formula:
αβ = c/a
In this equation, c = 4, substituting these values:
αβ = 4/2 = 2
Now, let's find the equation with roots α² and β². The sum of the roots (α² + β²) can be expressed as:
(α + β)² - 2αβ
Substituting the values we found earlier:
(α + β)² - 2αβ = (7/2)² - 2(2) = 49/4 - 4 = 49/4 - 16/4 = 33/4
Therefore, the equation whose roots are α² and β² is:
x² - (α + β)x + αβ = 0
x² - (7/2)x + 2 = 0
2(b). To evaluate (2ab + b)² - (b - 2a)² without using a mathematical table or calculator, we can simplify the expression step by step.
(2ab + b)² - (b - 2a)² can be written as [(2ab + b) + (b - 2a)] [(2ab + b) - (b - 2a)].
Let's simplify the first part:
(2ab + b) + (b - 2a) = 2ab + b + b - 2a = 2ab + 2b - 2a
Now, let's simplify the second part:
(2ab + b) - (b - 2a) = 2ab + b - b + 2a = 2ab + 2a
Multiplying the simplified forms together:
[(2ab + b) + (b - 2a)] [(2ab + b) - (b - 2a)] = (2ab + 2b - 2a)(2ab + 2a)
Expanding the product:
(2ab*2ab + 2ab*2a + 2b*-2a + 2b*2ab + 2b*2a - 2a*2a) = 4a²b² + 4a²b - 4ab² + 4ab + 4ab + 4a² - 4a²
Simplifying further:
4a²b² + 8a²b + 4ab = 4ab(2ab + 2) + 4a²b²
Therefore, (2ab + b)² - (b - 2a)² simplifies to 4ab(2ab + 2) + 4a²b².