**Middle School Number Sense Lesson 88: Discriminant and Roots of a Polynomial**

Today's lesson is about the most common middle school concept we've covered for almost 3 months. It has shown up

**15 times**so far this year, with a median placement at

**question # 74**.

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We should probably start with some definitions...

- The
**roots**of a polynomial are its solutions (or zeros)--the inputs which produce an output of**0**. - The
**discriminant**is a function of the polynomial's coefficients. When written as

**ax**,

^{2}+ bx + c = 0**discriminant** is:

**b**

^{2}- 4ac- The polynomial has
**2 real roots**if the discriminant is**positive**, - The polynomial has
**1 real (double) root**if the discriminant is**zero**, and - The polynomial has
**0 real roots**if the discriminant is**negative**.

Also grouped into this concept is the occasional problem where you are asked to identify a specific root. One way to do this is to factor the polynomial--which is covered by Khan Academy

**here**.

**Example 1:**

**The discriminant of 2x**

^{2}+ x - 3 = 0 is ___1. In this polynomial, a = 2, b = 1, and c = -3.

2. The discriminant is b

^{2}- 4ac, which is 1

^{2}- 4(2)(-3), which is 1 - 8(-3), which is 1 - (-24), which is

**25**.

**Example 2:**

**The discriminant of x**

^{2}+ 9x + 20 = 0 is ___1. a = 1, b = 9, and c = 20

2. b

^{2}- 4ac = 9

^{2}- 4(1)(20) = 81 - 80 =

**1**

**Example 3:**

**The discriminant of 9x**

^{2}- 1 = 0 is ___1. a = 9, b = 0, c = -1

2. b

^{2}- 4ac = 0

^{2}- 4(9)(-1) = 0 - 36(-1) = 0 + 36 =

**36**

**Example 4:**

**f(x) = x**

^{2}- 6x + 11 has how many real roots? ___1. Let's find the discriminant first. a = 1, b = -6, c = 11

2. b

^{2}- 4ac = (-6)

^{2}- 4(1)(11) = 36 - 44 =

**-8**

3. Since the discriminant is negative, the polynomial has

**0**real roots

**Example 5:**

**If f(x) = ax**

^{2}+ 12x + 4 has one distinct real root, then a = ___1. We need to set our discriminant equal to zero. a = ?, b = 12, and c = 4.

2. b

^{2}- 4ac = 0 --> 12

^{2}- 4(a)(4) = 0 --> 144 - 16a = 0

3. 144 = 16a, so a = 144/16 =

**9**

**Example 6:**

**The largest solution of x**

^{2}- 7x - 18 = 0 is ___1. We need to factor this polynomial:

**(x - 9)(x + 2) = 0**

2. Setting each binomial equal to 0, we get

**x - 9 = 0**and

**x + 2 = 0**

3. The roots are

**9**and

**-2**, and the larger of these is

**9**

**Example 7:**

**The smaller of the roots of x**

^{2}- 18x - 40 = 0 is ___1. Factoring, we get

**(x - 20)(x + 2) = 0**

2. Therefore

**x - 20 = 0**and

**x + 2 = 0**

3. The roots are

**20**and

**-2**, and the smaller of these is

**-2**

**Up Next for Middle School: SubFracDiff**