## Quadratics

### 7.1 The Discriminant

As we saw in part 2, the general quadratic formula follows from the box method:

**IF \(ax^{2}+bx+c=0\) THEN \(x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\).**

The term under the square root in the quadratic formula. \(b^{2}-4ac\), is called the *discriminant* of the quadratic expression .

Some curricula place emphasis on its study and have students note the following:

• If \(b^{2}-4ac\) is negative, then the quadratic has no real solutions. (One cannot compute the square root of a negative value.) Its graph does not cross the \(x\)-axis.

• If \(b^{2}-4ac\) equals \(0\), then the quadratic has precisely one solution. (Zero is the only number with precisely one square root.) Its graph just touches the \(x\)-axis.

• If \(b^{2}-4ac\) is positive, then the quadratic has two real solutions. (There are two square roots to a positive quantity.) Its graph crosses the \(x\)-axis twice.

If speed is important and memorization of formulas is considered par for the course, then one might indeed want to highlight the role of the discriminant. But as we saw in part 4, all questions about the count of times a quadratic crosses the horizontal axis can be answered with conceptual ease simply by making quick sketches and using common sense to fill in the details of those sketches.

I focus on the power of conceptual understand as the base of this course. (And placed all my commentary of the discriminant as OPTIONAL reading in part 2 of this course.)

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