X-intercept: y = 0
y-intercept: x = 0
3x + 4y = 0
x-intercept: subtitute y = 0
3x + 4 · 0 = 0
3x = 0
x = 0 → (0; 0)
y-intercept: subtitute x = 0
3 · 0 + 4y = 0
4y = 0
y = 0 → (0; 0)
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Suppose we wish to determine whether or not two given polynomials with complex coefficients have a common root. Given two first-degree polynomials a0 + a1x and b0 + b1x, we seek a single value of x such that
Solving each of these equations for x we get x = -a0/a1 and x = -b0/b1 respectively, so in order for both equations to be satisfied simultaneously we must have a0/a1 = b0/b1, which can also be written as a0b1 - a1b0 = 0. Formally we can regard this system as two linear equations in the two quantities x0 and x1, and write them in matrix form as
Hence a non-trivial solution requires the vanishing of the determinant of the coefficient matrix, which again gives a0b1 - a1b0 = 0.
Now consider two polynomials of degree 2. In this case we seek a single value of x such that
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