Quadratics in the xb plane

by Michael Walliser

Consider the equation **x ^{2} + bx + 1 = 0** (of the standard quadratic form

Now if we take an arbitrary value of **b**, for instance **b = 3**, and overlay this equation on the graph, we get a line parallel to the **x-axis**, as shown below. Anywhere this line intersects our original curve corresponds to the roots of the original equation for that value of **b**.

By observing the graph, it is easy to see that for all values **b > 2**, the line intersects the curve at two points to the left of the **b-axis**. Thus, the equation has two negative real roots. At **b = 2**, the line is tangent to the curve at a point to the left of the **b-axis**, so there is one negative real root. For values of **b** such that **-2 < b < 2**, the line does not intersect the curve, hence there are no real roots. When **b = -2**, the line is tangent to the curve at a point to the right of the **b-axis**, so there is one positive real root. For all values **b < -2**, the line intersects the curve at two points to the right of the **b-axis**. Thus, the original equation has two positive real roots.

Let's now consider the case for different values of **c**. The graph below shows **x ^{2} + bx + c = 0** in the

Notice that for each **c > 0**, there is a unique positive value, let's call it** u**, which we can use to define the key characteristics of the curve. When **b > u**, there are two negative real roots. When **b = u**, there is one negative root. When **-u < b < u**, there are no real roots. When **b = -u**, there is one positive root. Finally, when **b < -u**, there are two positive roots. From observation, it looks like **u** is equal to two times the square root of **c**. Let's see if this is indeed the case.

If **b** is greater than twice the square root of **c**, then **b ^{2}** is greater than

If **b** is equal to twice the square root of **c**, then **x ^{2} + bx + c** can be factored into

If **b** falls in the range **-2sqrt(c) < b < 2sqrt(c)**, then **b ^{2} < 4ac**, so from the quadratic formula, there are no real roots.

If **b** equals -2 times the square root of **c**, then **x ^{2} + bx + c** can be factored into

If **b** is less than -2 times the square root of **c**, then **b ^{2}** is greater than

So our postulate holds for all five cases when** c > 0**.

Now look at the cases where **c < 0**. We see that for every value of **b**, there is one positive real root and one negative real root. It looks also as though when **b = 0**, the roots are **+/- sqrt(-c)**. Is this true?

Well, when **b = 0**, then **x ^{2} + bx + c = 0** becomes simply

The final case to look at is **c = 0**. It would appear as though the roots are along the line **b = -x**. In other words, **x ^{2} + bx** is equal to zero when