' d ' is.
x + y + z = 0 x = a - b, y = b - c, z = c - a, therefore a - b + b - c + c - a = ? a - a + b - b + c - c = 0
B,C,c,D,E,H,I,K,l,M,N,O,T,U,V,v,W,w,X,x,Y
The next term is ' D '.
w=x-y w+y=x
d
' d ' is.
D
x + y + z = 0 x = a - b, y = b - c, z = c - a, therefore a - b + b - c + c - a = ? a - a + b - b + c - c = 0
The next term is ' d ' .
tan (A-B) + tan (B-C) + tan (C-A)=0 tan (A-B) + tan (B-C) - tan (A-C)=0 tan (A-B) + tan (B-C) = tan (A-C) (A-B) + (B-C) = A-C So we can solve tan (A-B) + tan (B-C) = tan (A-C) by first solving tan x + tan y = tan (x+y) and then substituting x = A-B and y = B-C. tan (x+y) = (tan x + tan y)/(1 - tan x tan y) So tan x + tan y = (tan x + tan y)/(1 - tan x tan y) (tan x + tan y)tan x tan y = 0 So, tan x = 0 or tan y = 0 or tan x = - tan y tan(A-B) = 0 or tan(B-C) = 0 or tan(A-B) = - tan(B-C) tan(A-B) = 0 or tan(B-C) = 0 or tan(A-B) = tan(C-B) A, B and C are all angles of a triangle, so are all in the range (0, pi). So A-B and B-C are in the range (- pi, pi). At this point I sketched a graph of y = tan x (- pi < x < pi) By inspection I can see that: A-B = 0 or B-C = 0 or A-B = C-B or A-B = C-B +/- pi A = B or B = C or A = C or A = C +/- pi But A and C are both in the range (0, pi) so A = C +/- pi has no solution So A = B or B = C or A = C A triangle ABC has the property that tan (A-B) + tan (B-C) + tan (C-A)=0 if and only if it is isosceles (or equilateral).
B,C,c,D,E,H,I,K,l,M,N,O,T,U,V,v,W,w,X,x,Y
The next term is ' D '.
404
It depends upon the equation that you are trying to solve. for example if you had x² + y² - 4x - 6y = 3 completing the square (in both x and y) would give you (x - 2)² - 4 + (y - 3)² - 9 = 3 at which point you could collect the numbers together on the right to give you (x - 2)² + (y - 3)² = 16 = 4² and then "read off" the centre of the circle at (2, 3) and its radius of 4. However, I suspect that you are asking regarding a quadratic, in which case you have got to something like (x - w)² - r = 0 Just proceed as normal with an equation to solve: (x - w)² - r = 0 → (x - w)² = r → x - w = ±√r → x = w ±√r ------------------------------------------------------------------ For a quadratic of the form ax² + bx +c this gives: ax² + bx + c = 0 → x² +(b/a)x + (c/a) = 0 completing the square: → (x + (b/2a))² - (b/2a)² + c/a = 0 → (x + (b/2a))² = b²/4a² - 4ac/4a² → (x + (b/2a))² = (b² - 4ac)/4a² → x + (b/2a) = ±√((b² - 4ac)/4a²) → x = -(b/2a) ±(√(b² - 4ac))/2a → x = (-b ±√(b² - 4ac))/2a which you may recognise as the formula for solving a quadratic equation.
A quadratic involving x and y is usually in the form 'y = ax2 + bx + c'. This form is y in terms of x, so we must rearrange it. y = ax2 + bx + c y/a = x2 + bx/a + c/a y/a = x2 + bx/a + d + e, where c/a = d + e, e = (b/a)2 y/a - e = x2 + bx/a + d y/a -e = (x + b/a)2 √(y/a - e) = x + b/a √(y/a - e) - b/a = x
The 'x' coordinate of B is the average of the 'x' coordinates of A and C. The 'y' coordinate of B is the average of the 'y' coordinates of A and C.