Question

1.What is the shape of the graph of any quadratic function?

2. What kind of solution(s) does ax^2 - bx + c = 0 have if b^2 - 4ac < 0?
a.two real solutions
b.one real solution
c.one complex solutions
d.two complex solutions

Answer 1

Inverted parabola

Two complex solution

Explanation:

any quadratic equation can be expressed in terms of ax^2 + bx + c = 0

It has two solutions which can be found using formula
`X =( -b + √(b^2 -4ac) )/2a \\or \ ( -b - √(b^2 -4ac) )/2a`

We know that for any value inside the square root function

if it is positive its solution will be real number

for example
`√(9)` we can have two real solution -3 and 3

if it is negative its solution will be complex number

for example
`√(-9)`we can have two complex solution

expressed in term of a + ib or a - ib

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Given in the problem

we have ax^2 - bx + c

so its solution will be

`X =( -(-b) + √(b^2 -4ac) )/2a \\or \ ( -(-b) - √(b^2 -4ac) )/2a\\\\X =( b + √(b^2 -4ac) )/2a \\or \ ( b - √(b^2 -4ac) )/2a\\`

Since in this expression it is given in problem statement that b^2 - 4ac < 0 which will be a negative number and for negative number, its square root is complex number.

also X has two values as given above it will have two solution

hence it will have two complex solution based on above discussion.

Also for any quadratic equation shape of its curve if plotted on coordinate plane is parabolic.

If it has complex solution then curve of such expression will not pass through X axis