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\( x^2 \)
\( = 2 \text{ roots} \)

Fundamental Theorem
of Algebra

Discover why every quadratic equation always has exactly two roots when we count with multiplicity in the complex numbers.

Algebra 2

What Is a Root of a Polynomial?

A root (or zero) of a polynomial \( p(x) \) is any value \( r \) such that \( p(r) = 0 \). Graphically, roots are the x-intercepts where the curve crosses or touches the x-axis.

Real vs Complex Numbers

Real Numbers
All numbers on the number line: integers, fractions, and irrationals like \( \sqrt{2} \) and \( \pi \).
Complex Numbers
Numbers of the form \( a + bi \), where \( i = \sqrt{-1} \). Every real number is complex with \( b = 0 \).

The Discriminant Decides

For \( ax^2 + bx + c = 0 \), the expression under the square root in the quadratic formula is called the discriminant:

\( \Delta > 0 \)
Two distinct real roots
\( \Delta = 0 \)
One repeated real root (multiplicity 2)
\( \Delta < 0 \)
Two distinct complex conjugate roots

Three Cases, Visually

\( \Delta > 0 \)
 2 real roots
\( \Delta = 0 \)
 1 repeated root
\( \Delta < 0 \)
 2 complex roots

The Fundamental Theorem of Algebra

For a quadratic \( ax^2 + bx + c \) (degree \( n = 2 \)), this guarantees exactly 2 roots in the complex numbers, no matter what \( a \), \( b \), and \( c \) are (as long as \( a \neq 0 \)).

Proving It for Quadratics

1
Start with \( ax^2 + bx + c = 0 \) where \( a \neq 0 \).
2
Apply the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
3
If \( \Delta \geq 0 \): \( \sqrt{\Delta} \) is real, giving two real roots (possibly equal).
4
If \( \Delta < 0 \): \( \sqrt{\Delta} = i\sqrt{|\Delta|} \), giving two complex conjugate roots \( \frac{-b}{2a} \pm \frac{\sqrt{|\Delta|}}{2a}\,i \).
5
Conclusion: In every case, there are exactly 2 roots in \( \mathbb{C} \). ✓

Roots and Factored Form

If \( r_1 \) and \( r_2 \) are the two roots, then the quadratic factors completely over \( \mathbb{C} \):

Vieta's Formulas
\( r_1 + r_2 = -\frac{b}{a} \) and \( r_1 \cdot r_2 = \frac{c}{a} \)
Complete Factorization
Every degree-2 polynomial splits into two linear factors over \( \mathbb{C} \).

Real Roots vs Complex Roots

Real Roots Complex Roots
Discriminant \( \Delta \geq 0 \) \( \Delta < 0 \)
Graph Behavior Parabola crosses or touches x-axis Parabola stays entirely above or below x-axis
Root Form Pure real numbers Conjugate pairs: \( a \pm bi \)
Count (with multiplicity) Always 2 Always 2

What Is Multiplicity?

When we say "counted with multiplicity," we mean that a repeated root is counted as many times as it appears as a factor.

\( x^2 - 4 = 0 \)
Roots: \( x = 2 \) and \( x = -2 \). Two distinct roots, each multiplicity 1.
\( x^2 - 6x + 9 = 0 \)
Factors as \( (x - 3)^2 = 0 \). One root \( x = 3 \) with multiplicity 2.

Conjugate Pairs Always Come Together

When a quadratic has real coefficients and a negative discriminant, the two complex roots are always conjugates of each other:

This is because the \( \pm \) in the quadratic formula produces both \( +i\sqrt{|\Delta|} \) and \( -i\sqrt{|\Delta|} \). The imaginary parts are equal in magnitude but opposite in sign.

How to Find All Roots

1
Identify \( a \), \( b \), and \( c \) from \( ax^2 + bx + c = 0 \).
2
Compute the discriminant: \( \Delta = b^2 - 4ac \).
3
If \( \Delta \geq 0 \): use \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \) to get two real roots.
4
If \( \Delta < 0 \): write \( \sqrt{\Delta} = i\sqrt{|\Delta|} \) and get two complex conjugate roots.
5
Verify: confirm both roots satisfy the original equation, and the total is 2 (with multiplicity).

Example 1: Two Real Roots

Solve \( x^2 - 5x + 6 = 0 \) and verify the Fundamental Theorem.

1
Identify: \( a = 1 \), \( b = -5 \), \( c = 6 \).
2
Discriminant: \( \Delta = (-5)^2 - 4(1)(6) = 25 - 24 = \)\( 1 \). Since \( \Delta > 0 \), two distinct real roots.
3
Solve: \( x = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2} \), so \( x = \)\( 3 \) or \( x = \)\( 2 \).

Example 2: Repeated Root

Solve \( 4x^2 + 12x + 9 = 0 \) and verify the Fundamental Theorem.

1
Identify: \( a = 4 \), \( b = 12 \), \( c = 9 \).
2
Discriminant: \( \Delta = (12)^2 - 4(4)(9) = 144 - 144 = \)\( 0 \). One repeated root.
3
Solve: \( x = \frac{-12}{2(4)} = -\frac{3}{2} \). This root has multiplicity 2.

Example 3: Complex Roots

Solve \( x^2 + 2x + 5 = 0 \) and verify the Fundamental Theorem.

1
Identify: \( a = 1 \), \( b = 2 \), \( c = 5 \).
2
Discriminant: \( \Delta = (2)^2 - 4(1)(5) = 4 - 20 = \)\( -16 \). Since \( \Delta < 0 \), two complex roots.
3
Solve: \( x = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i \).

Common Mistakes to Avoid

✕ "No Solution" for Negative Discriminant
Saying "no roots" when \( \Delta < 0 \). There are no real roots, but there are always two complex roots.
✕ Forgetting Multiplicity
Counting a repeated root only once. The root \( x = 3 \) in \( (x - 3)^2 = 0 \) has multiplicity 2, not 1.
✕ Sign Error with \( i \)
Writing \( \sqrt{-16} = -4 \) instead of \( \sqrt{-16} = 4i \). The square root of a negative number uses \( i \), not a negative sign.

Summary: All Three Cases

Discriminant Nature of Roots Example Total Roots
\( \Delta > 0 \) 2 distinct real \( x^2 - 5x + 6 = 0 \) 2
\( \Delta = 0 \) 1 repeated real \( x^2 - 6x + 9 = 0 \) 2
\( \Delta < 0 \) 2 complex conjugates \( x^2 + 2x + 5 = 0 \) 2

Question 1 of 5

According to the Fundamental Theorem of Algebra, how many roots does the polynomial \( 3x^2 + 7x - 2 \) have in the complex numbers?

Question 2 of 5

What is the discriminant of \( x^2 + 4x + 13 = 0 \), and what does it tell us about the roots?

Question 3 of 5

The quadratic \( x^2 - 10x + 25 = 0 \) has the root \( x = 5 \). How many roots does it have counted with multiplicity?

Question 4 of 5

If one root of \( x^2 + 6x + 13 = 0 \) is \( x = -3 + 2i \), what is the other root?

Question 5 of 5

Which statement best describes why the Fundamental Theorem of Algebra is true for quadratics?