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X-intercepts and Y-intercept of biquadratic equations


When we speak about curve plotted from a function, we sometimes are interested to know where will the curve intersect x-axis or y-axis. Points where the curve intersect with x-axis is called x-intercept, while point where the curve intersect with y-axis is called y-intercept. A biquadratic equation can have up to four x-intercept and will always have one y-intercept.




Using Orimath Biquadratic Equation to find the x-intercepts and y-intercept of a biquadratic equation is a simple affair. All you need to do is to check the check boxes titled "x-intercepts" and "y-intercept", depending on which kind of intercept you are looking for. Then click as HTML button to get the result.


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Answer by Orimath Quadratic Solver :
Question Number 3 :
For this function y(x) = x4 - 5x2 + 4 , answer the following questions :

Answer Number 3 :
The equation x4 - 5x2 + 4 = 0 is already in ax4+bx2+c=0 form.
As the value is already arranged in ax4+bx2+c=0 form, we get the value of a = 1, b = -5, c = 4.

For the curve y = x4 - 5x2 + 4 to intersect with x-axis,
We have to remember that x-axis itself is a line with equation y = 0
A way to find value of x and y that match both equations is by subtituting y with 0
So we have to solve : x4 - 5x2 + 4 = 0
We will use completing the square to solve the equation
x4 - 5x2 + 4 = 0 ,divide both side with 1
So we get x4 - 5x2 + 4 = 0 ,
We know that the coefficient of x is -5
We have to use the fact that ( x + q )2 = x2 + 2qx + q2 , and assume that q = -5/2 = -2.5
So we have make the equation into x4 - 5x2 + 6.25 - 2.25 = 0
And it is the same with ( x2 - 2.5 )2 - 2.25 = 0
And it is the same with (( x2 - 2.5 ) - 1.5 ) (( x2 - 2.5 ) + 1.5 ) = 0
And it is the same with ( x2 - 2.5 - 1.5 ) ( x2 - 2.5 + 1.5 ) = 0
Just add up the constants in each brackets, and we get ( x2 - 4 ) ( x2 - 1 ) = 0
The equation x4 - 5x2 + 4 = 0 , have four roots :
Root 1 : x1 =      x12   = 4   = 2
Root 2 : x2 =      x22   = 1   = 1
Root 3 : x3 = -     x12   = -4   = -2
Root 4 : x4 = -     x22   = -1   = -1
Which means that the function y(x) = x4 - 5x2 + 4 :
Have four x-intercept in real Cartesian coordinate, they are :
( x , y ) = ( 2 , 0 )
( x , y ) = ( 1 , 0 )
( x , y ) = ( -2 , 0 )
( x , y ) = ( -1 , 0 )
The coordinate where the curve y = x4 - 5x2 + 4 intersect with y-axis,
will always be in the form of ( x , y ) = ( 0 ,y )
A way to find value of x and y that match both equations is by subtituting x with 0
So we get y = 1 (0)2 + -5 (0) + 4
Which make y = 4
So the function have y-intercept in ( x , y ) = ( 0 , 4 )

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