What does it mean when the roots of two quadratic equations are 'simultaneously real'?

Question

Does it mean the equations have equal roots? Or the equations have common solution(s)?

I'll give the question as it is. This simultaneously real part is what I don't get.

If the roots of the equations ax^2 + 2bx + c = 0
and bx^2 - [2x * sqrt (ac)] + b = 0

are simultaneously real, the prove that b^2 = ac

Answer 1

this means the both eqautions have real solutions

If the roots of the equations ax^2 + 2bx + c = 0

then discriminant >or (2b)^2 - 4ac >= 0

or b^2>= 4ac ..1

bx^2 - [2x * sqrt (ac)] + b = 0
has real roots

this means B^2-4AC >=0 (A =b, C=b and B= 2sqrt(ac)
or (4ac) ^2-4b^2 >= 0
or ac >= b^2 ..2

for both to have real roots both 1a and 2 should hold

that is
b^2 = ac

proved

Answer 2

indexed here are 2 elementary the thank you to think of approximately this. at the beginning, we can truthfully form an equation (not a quadratic) that has roots equivalent to the reciprocal of the roots of the given quadratic, merely by means of substituting a million/x somewhat of x: a(a million/x)^2 + b(a million/x) + c = 0 think of approximately it. If some x satisfies this, then a million/x will fulfill the unique equation, and vice-versa. Now, we merely multiply by means of by means of the easy denominator x^2: a + bx + cx^2 = 0 cx^2 + bx + a = 0 the only situation shall we've is that if between the roots is 0, that's resembling asserting c = 0 (merely by means of substituting x = 0 into the unique equation). So, this works on the circumstance that c is non-0. any different way is to evaluate the coefficients as purposes of the roots. all of us understand that, if p and q are the roots: -b/a = p + q c/a = pq we ought to locate a million/p + a million/q and (a million/p)(a million/q). we've: (a million/p)(a million/q) = a million/(pq) = a/c a million/p + a million/q = q/(pq) + p/(pq) = (p + q) / (pq) = (-b/a) / (c/a) = -b/c for this reason, one such quadratic equation could be: x^2 + (b/c)x + a/c = cx^2 + bx + a desire that facilitates!

Answer 3

Your question says for TWO quadratic EQUATIONS. That simultaneity probably means they have the same discriminant which is also greater than or equal to zero.

An example might be:

x^2 - 10x + 9 = 0 a=1, b=-10, c=9, b^2 - 4ac=64
3x^2 + 10x +3 = 0 a=c=3, b=10, b^2 - 4ac=64

Real roots for both, but they have different roots.

Answer 4

hay!
u just solve the 2 equations simeltaneously then get the roots
both roots will be real as the resulting quadratic equation has a discriminante >or= to 0
hope i helped

Answer 5

eqoations have only 1 solution