Algebra 2 Flashcards: Fundamental Theorem Of Algebra For Quadratics

What this deck covers

This deck focuses on Fundamental Theorem Of Algebra For Quadratics, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find the roots of $7x^2-1=0$.

Answer: $x=\pm\frac{1}{\sqrt{7}}$. Solving $7x^2=1$ gives $x^2=\frac{1}{7}$.

Flashcard 2: What is the definition of the imaginary unit $i$?

Answer: $i^2=-1$. Fundamental definition of the imaginary unit.

Flashcard 3: Find the complex roots of $x^2-8x+20=0$.

Answer: $x=4\pm 2i$. Using quadratic formula with $\Delta=-16<0$.

Flashcard 4: What is the conjugate of a complex number $a+bi$?

Answer: $a-bi$. Complex conjugate changes the sign of the imaginary part.

Flashcard 5: Find the complex roots of $x^2-2x+2=0$.

Answer: $x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.

Flashcard 6: What does the Complex Conjugate Root Theorem say for polynomials with real coefficients?

Answer: If $a+bi$ is a root, then $a-bi$ is a root. Real coefficients force complex roots to come in conjugate pairs.

Flashcard 7: What is the conjugate of a complex number $a+bi$?

Answer: $a-bi$. Complex conjugate changes the sign of the imaginary part.

Flashcard 8: What does the Factor Theorem state about $r$ and the factor $(x-r)$?

Answer: $p(r)=0\iff(x-r)$ is a factor of $p(x)$. Fundamental connection between roots and linear factors.

Flashcard 9: What is the discriminant of $ax^2+bx+c$?

Answer: $\Delta=b^2-4ac$. The discriminant determines the nature of the roots.

Flashcard 10: Which discriminant condition gives two distinct real roots for $ax^2+bx+c=0$?

Answer: $\Delta>0$. Positive discriminant means the square root yields two real values.

Flashcard 11: Which discriminant condition gives exactly one real root (a repeated root) for $ax^2+bx+c=0$?

Answer: $\Delta=0$. Zero discriminant makes the $\pm$ term disappear, giving one solution.

Flashcard 12: Which discriminant condition gives two nonreal complex roots for $ax^2+bx+c=0$?

Answer: $\Delta<0$. Negative discriminant requires imaginary unit for square root.

Flashcard 13: What is the definition of the imaginary unit $i$?

Answer: $i^2=-1$. Fundamental definition of the imaginary unit.

Flashcard 14: How do you rewrite $\sqrt{-k}$ for $k>0$ using $i$?

Answer: $\sqrt{-k}=i\sqrt{k}$. Standard method to express square roots of negative numbers.

Flashcard 15: What is the quadratic formula for the solutions of $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard formula derived from completing the square method.

Flashcard 16: What is the degree of a quadratic polynomial written as $ax^2+bx+c$ with $a\ne 0$?

Answer: Degree $2$. The highest power term $ax^2$ determines the degree is $2$.

Flashcard 17: What does it mean for a complex number to be a root of $p(x)$?

Answer: $p(r)=0$. A root makes the polynomial evaluate to zero.

Flashcard 18: What does the Fundamental Theorem of Algebra guarantee for any nonconstant polynomial with complex coefficients?

Answer: It has at least one complex root. FTA guarantees at least one complex root for any nonconstant polynomial.

Flashcard 19: What does the Fundamental Theorem of Algebra say about the number of complex roots of a degree $n$ polynomial (counting multiplicity)?

Answer: Exactly $n$ complex roots, counting multiplicity. FTA states degree $n$ polynomial has exactly $n$ complex roots when counting multiplicity.

Flashcard 20: What does the Fundamental Theorem of Algebra imply about the total number of complex zeros of a quadratic (counting multiplicity)?

Answer: It has $2$ complex zeros, counting multiplicity. Since degree is $2$, FTA guarantees exactly $2$ complex zeros.

Flashcard 21: Write the monic quadratic with roots $2+3i$ and $2-3i$.

Answer: $x^2-4x+13$. Using Vieta's formulas: sum=$4$, product=$13$.

Flashcard 22: Find the roots of $x^2-6x+13=0$ in $a\pm bi$ form.

Answer: $x=3\pm 2i$. Using quadratic formula with discriminant $\Delta=-16<0$.

Flashcard 23: Factor $x^2-2x+5$ over the complex numbers.

Answer: $(x-(1+2i))(x-(1-2i))$. Using roots $1+2i$ and $1-2i$ to form linear factors.

Flashcard 24: If a real-coefficient quadratic has root $-1+2i$, what is the other root?

Answer: $-1-2i$. Complex Conjugate Root Theorem for real coefficients.

Flashcard 25: Identify the conjugate of the complex root $3-5i$.

Answer: $3+5i$. Complex conjugate of $3-5i$ flips the imaginary sign.

Flashcard 26: Find the complex roots of $x^2-2x+2=0$.

Answer: $x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.

Flashcard 27: Write $x^2+6x+9$ in factored form showing multiplicity.

Answer: $(x+3)^2$. Perfect square trinomial factors as $(x+3)^2$.

Flashcard 28: What is the repeated root of $x^2+6x+9=0$?

Answer: $x=-3$. Since $\Delta=0$, there's one repeated root.

Flashcard 29: Find the discriminant $\Delta$ of $x^2+6x+9=0$.

Answer: $\Delta=0$. Using $\Delta=b^2-4ac$ with $a=1,b=6,c=9$.

Flashcard 30: Find the solutions of $2x^2+3x-2=0$.

Answer: $x=\frac{1}{2}$ and $x=-2$. Using quadratic formula with $\Delta=25>0$.