If α and β are the roots of equation x^2-8x +12=0, what is the value of α^2+β^2?
Answers
EXPLANATION.
α and β are the roots of the equation.
⇒ x² - 8x + 12 = 0.
As we know that,
Sum of the zeroes of the quadratic polynomial.
⇒ α + β = - b/a.
⇒ α + β = -(-8)/1 = 8.
Products of the zeroes of the quadratic polynomial.
⇒ αβ = c/a.
⇒ αβ = 12/1 = 12.
To find :
⇒ α² + β².
⇒ α² + β² = (α + β)² - 2αβ.
⇒ α² + β² = (8)² - 2(12).
⇒ α² + β² = 64 - 24.
⇒ α² + β² = 40.
MORE INFORMATION.
Transformation of equations.
Rules to form an equation whose roots are given in terms of another equation. Let given equation be.
α₀xⁿ + α₁xⁿ⁻¹ + . . . . . + αₙ₋₁x + αₙ = 0. - - - - - (1).
Rule = 1.
To form an equation whose roots are (k ≠ 0) times roots of the equation in (1) replace x by x/k in equation (1).
Rule = 2.
To form an equation whose roots are the negative of the roots in equation (1) replace x by - x in equation (1). Alternatively, change the sign of the coefficient of xⁿ⁻¹, xⁿ⁻³, xⁿ⁻⁵ . . . . . etc. in equation (1).
Rule = 3.
To form an equation whose roots are k more than the roots of the equation (1) replace x by x - k in equation (1).
Rule = 4.
To form an equation whose roots are reciprocals of the roots in equation (1) replace x by 1/x in equation (1) and then multiply both sides by xⁿ.
Rule = 5.
To form an equation whose roots are square of the roots of the equation (1) proceed as follows,
Step - 1.
Replace x by √x in equation (1).
Step - 2.
Collect all the terms involving √x on one side.
Step - 3.
Square both the sides and simplify.
For to form equation whose roots are square of the roots of x³ - 2x² - x + 2 = 0.
Replace x by √x to obtain.
x√x - 2x - √x + 2 = 0.
√x(x - 1) = 2(x - 1).
Squaring we get,
x(x - 1)² = 4(x - 1)².
(x - 4)(x² - 2x + 1) = 0. Or x³ - 6x² + 9x - 4 = 0.
Rule = 6.
To form an equation whose roots are cube of the roots of the equation in (1) proceed as follows,
Step - 1.
Replace x by x^(1/3).
Step - 2.
Collect all the terms involving x^(1/3) and x^(2/3) on one side.
Step - 3.
Cube both the sides and simplify.