Remember that the Quadratic Formula<span> solves "</span><span>ax2 + bx + c = 0</span><span>" for the values of </span>x<span>. Also remember that this means that you are trying to find the </span>x-intercepts of the graph. When the Formula gives you a negative inside the square root, you can now simplify that zero by using complex numbers. The answer you come up with is a valid "zero" or "root" or "solution" for "<span>ax2 + bx + c = 0</span><span>", because, if you plug it back into the quadratic, you'll get zero after you simplify. But you cannot graph a complex number on the </span>x,y<span>-plane. So this "solution to the equation" is not an </span>x-intercept. You can make this connection between the Quadratic Formula, complex numbers, and graphing:
<span><span><span>x2 – 2x – 3</span><span>x2 – 6x + 9</span><span>x2 + 3x + 3</span></span><span>a positive number inside the square rootzero inside the square roota negative number inside the square root</span><span>two real solutionsone (repeated) real solutiontwo complex solutions</span><span><span><span>two distinct </span>x-intercepts</span><span><span>one (repeated) </span>x-intercept</span><span><span>no </span>x-intercepts</span></span></span>
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<span>As an aside, you can graph complexes, but not in the </span>x,y<span>-plane. You need the "complex" plane. For the complex plane, the </span>x<span>-axis is where you plot the real part, and the </span>y<span>-axis is where you graph the imaginary part. For instance, you would plot the complex number </span><span>3 – 2i</span><span> like this:</span>
<span>This leads to an interesting fact: When you learned about regular ("real") numbers, you also learned about their order (this is what you show on the number line). But </span>x,y<span>-points don't come in any particular order. You can't say that one point "comes after" another point in the same way that you can say that one number comes after another number. For instance, you can't say that </span>(4, 5)<span> "comes after" </span>(4, 3)<span> in the way that you can say that </span>5<span> comes after </span>3<span>. Pretty much all you can do is compare "size", and, for complex numbers, "size" means "how far from the origin". To do this, you use the </span>Distance Formula, and compare which complexes are closer to or further from the origin. This "size" concept is called "the modulus". For instance, looking at our complex number plotted above, its modulus is computed by using the Distance Formula: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved
<span>Note that all points at this distance from the origin have the same modulus. All the points on the circle with radius </span>sqrt(13)<span> are viewed as being complex numbers having the same "size" as </span><span>3 – 2i</span><span>.</span>
