Fundamental Theorem of Algebra for Quadratics - Algebra 2
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Find the roots of $7x^2-1=0$.
Find the roots of $7x^2-1=0$.
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$x=\pm\frac{1}{\sqrt{7}}$. Solving $7x^2=1$ gives $x^2=\frac{1}{7}$.
$x=\pm\frac{1}{\sqrt{7}}$. Solving $7x^2=1$ gives $x^2=\frac{1}{7}$.
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What is the definition of the imaginary unit $i$?
What is the definition of the imaginary unit $i$?
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$i^2=-1$. Fundamental definition of the imaginary unit.
$i^2=-1$. Fundamental definition of the imaginary unit.
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Find the complex roots of $x^2-8x+20=0$.
Find the complex roots of $x^2-8x+20=0$.
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$x=4\pm 2i$. Using quadratic formula with $\Delta=-16<0$.
$x=4\pm 2i$. Using quadratic formula with $\Delta=-16<0$.
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What is the conjugate of a complex number $a+bi$?
What is the conjugate of a complex number $a+bi$?
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$a-bi$. Complex conjugate changes the sign of the imaginary part.
$a-bi$. Complex conjugate changes the sign of the imaginary part.
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Find the complex roots of $x^2-2x+2=0$.
Find the complex roots of $x^2-2x+2=0$.
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$x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.
$x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.
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What does the Complex Conjugate Root Theorem say for polynomials with real coefficients?
What does the Complex Conjugate Root Theorem say for polynomials with real coefficients?
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If $a+bi$ is a root, then $a-bi$ is a root. Real coefficients force complex roots to come in conjugate pairs.
If $a+bi$ is a root, then $a-bi$ is a root. Real coefficients force complex roots to come in conjugate pairs.
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What is the conjugate of a complex number $a+bi$?
What is the conjugate of a complex number $a+bi$?
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$a-bi$. Complex conjugate changes the sign of the imaginary part.
$a-bi$. Complex conjugate changes the sign of the imaginary part.
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What does the Factor Theorem state about $r$ and the factor $(x-r)$?
What does the Factor Theorem state about $r$ and the factor $(x-r)$?
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$p(r)=0\iff(x-r)$ is a factor of $p(x)$. Fundamental connection between roots and linear factors.
$p(r)=0\iff(x-r)$ is a factor of $p(x)$. Fundamental connection between roots and linear factors.
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What is the discriminant of $ax^2+bx+c$?
What is the discriminant of $ax^2+bx+c$?
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$\Delta=b^2-4ac$. The discriminant determines the nature of the roots.
$\Delta=b^2-4ac$. The discriminant determines the nature of the roots.
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Which discriminant condition gives two distinct real roots for $ax^2+bx+c=0$?
Which discriminant condition gives two distinct real roots for $ax^2+bx+c=0$?
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$\Delta>0$. Positive discriminant means the square root yields two real values.
$\Delta>0$. Positive discriminant means the square root yields two real values.
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Which discriminant condition gives exactly one real root (a repeated root) for $ax^2+bx+c=0$?
Which discriminant condition gives exactly one real root (a repeated root) for $ax^2+bx+c=0$?
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$\Delta=0$. Zero discriminant makes the $\pm$ term disappear, giving one solution.
$\Delta=0$. Zero discriminant makes the $\pm$ term disappear, giving one solution.
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Which discriminant condition gives two nonreal complex roots for $ax^2+bx+c=0$?
Which discriminant condition gives two nonreal complex roots for $ax^2+bx+c=0$?
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$\Delta<0$. Negative discriminant requires imaginary unit for square root.
$\Delta<0$. Negative discriminant requires imaginary unit for square root.
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What is the definition of the imaginary unit $i$?
What is the definition of the imaginary unit $i$?
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$i^2=-1$. Fundamental definition of the imaginary unit.
$i^2=-1$. Fundamental definition of the imaginary unit.
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How do you rewrite $\sqrt{-k}$ for $k>0$ using $i$?
How do you rewrite $\sqrt{-k}$ for $k>0$ using $i$?
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$\sqrt{-k}=i\sqrt{k}$. Standard method to express square roots of negative numbers.
$\sqrt{-k}=i\sqrt{k}$. Standard method to express square roots of negative numbers.
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What is the quadratic formula for the solutions of $ax^2+bx+c=0$?
What is the quadratic formula for the solutions of $ax^2+bx+c=0$?
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$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard formula derived from completing the square method.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard formula derived from completing the square method.
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What is the degree of a quadratic polynomial written as $ax^2+bx+c$ with $a\ne 0$?
What is the degree of a quadratic polynomial written as $ax^2+bx+c$ with $a\ne 0$?
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Degree $2$. The highest power term $ax^2$ determines the degree is $2$.
Degree $2$. The highest power term $ax^2$ determines the degree is $2$.
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What does it mean for a complex number to be a root of $p(x)$?
What does it mean for a complex number to be a root of $p(x)$?
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$p(r)=0$. A root makes the polynomial evaluate to zero.
$p(r)=0$. A root makes the polynomial evaluate to zero.
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What does the Fundamental Theorem of Algebra guarantee for any nonconstant polynomial with complex coefficients?
What does the Fundamental Theorem of Algebra guarantee for any nonconstant polynomial with complex coefficients?
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It has at least one complex root. FTA guarantees at least one complex root for any nonconstant polynomial.
It has at least one complex root. FTA guarantees at least one complex root for any nonconstant polynomial.
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What does the Fundamental Theorem of Algebra say about the number of complex roots of a degree $n$ polynomial (counting multiplicity)?
What does the Fundamental Theorem of Algebra say about the number of complex roots of a degree $n$ polynomial (counting multiplicity)?
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Exactly $n$ complex roots, counting multiplicity. FTA states degree $n$ polynomial has exactly $n$ complex roots when counting multiplicity.
Exactly $n$ complex roots, counting multiplicity. FTA states degree $n$ polynomial has exactly $n$ complex roots when counting multiplicity.
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What does the Fundamental Theorem of Algebra imply about the total number of complex zeros of a quadratic (counting multiplicity)?
What does the Fundamental Theorem of Algebra imply about the total number of complex zeros of a quadratic (counting multiplicity)?
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It has $2$ complex zeros, counting multiplicity. Since degree is $2$, FTA guarantees exactly $2$ complex zeros.
It has $2$ complex zeros, counting multiplicity. Since degree is $2$, FTA guarantees exactly $2$ complex zeros.
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Write the monic quadratic with roots $2+3i$ and $2-3i$.
Write the monic quadratic with roots $2+3i$ and $2-3i$.
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$x^2-4x+13$. Using Vieta's formulas: sum=$4$, product=$13$.
$x^2-4x+13$. Using Vieta's formulas: sum=$4$, product=$13$.
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Find the roots of $x^2-6x+13=0$ in $a\pm bi$ form.
Find the roots of $x^2-6x+13=0$ in $a\pm bi$ form.
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$x=3\pm 2i$. Using quadratic formula with discriminant $\Delta=-16<0$.
$x=3\pm 2i$. Using quadratic formula with discriminant $\Delta=-16<0$.
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Factor $x^2-2x+5$ over the complex numbers.
Factor $x^2-2x+5$ over the complex numbers.
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$(x-(1+2i))(x-(1-2i))$. Using roots $1+2i$ and $1-2i$ to form linear factors.
$(x-(1+2i))(x-(1-2i))$. Using roots $1+2i$ and $1-2i$ to form linear factors.
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If a real-coefficient quadratic has root $-1+2i$, what is the other root?
If a real-coefficient quadratic has root $-1+2i$, what is the other root?
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$-1-2i$. Complex Conjugate Root Theorem for real coefficients.
$-1-2i$. Complex Conjugate Root Theorem for real coefficients.
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Identify the conjugate of the complex root $3-5i$.
Identify the conjugate of the complex root $3-5i$.
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$3+5i$. Complex conjugate of $3-5i$ flips the imaginary sign.
$3+5i$. Complex conjugate of $3-5i$ flips the imaginary sign.
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Find the complex roots of $x^2-2x+2=0$.
Find the complex roots of $x^2-2x+2=0$.
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$x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.
$x=1\pm i$. Using quadratic formula with $\Delta=-4<0$.
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Write $x^2+6x+9$ in factored form showing multiplicity.
Write $x^2+6x+9$ in factored form showing multiplicity.
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$(x+3)^2$. Perfect square trinomial factors as $(x+3)^2$.
$(x+3)^2$. Perfect square trinomial factors as $(x+3)^2$.
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What is the repeated root of $x^2+6x+9=0$?
What is the repeated root of $x^2+6x+9=0$?
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$x=-3$. Since $\Delta=0$, there's one repeated root.
$x=-3$. Since $\Delta=0$, there's one repeated root.
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Find the discriminant $\Delta$ of $x^2+6x+9=0$.
Find the discriminant $\Delta$ of $x^2+6x+9=0$.
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$\Delta=0$. Using $\Delta=b^2-4ac$ with $a=1,b=6,c=9$.
$\Delta=0$. Using $\Delta=b^2-4ac$ with $a=1,b=6,c=9$.
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Find the solutions of $2x^2+3x-2=0$.
Find the solutions of $2x^2+3x-2=0$.
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$x=\frac{1}{2}$ and $x=-2$. Using quadratic formula with $\Delta=25>0$.
$x=\frac{1}{2}$ and $x=-2$. Using quadratic formula with $\Delta=25>0$.
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