Solve Quadratics by Multiple Methods - Algebra
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What does $\Delta<0$ tell you about the solutions of a quadratic?
What does $\Delta<0$ tell you about the solutions of a quadratic?
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Two complex (nonreal) solutions. Negative discriminant means roots involve imaginary numbers.
Two complex (nonreal) solutions. Negative discriminant means roots involve imaginary numbers.
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What is the imaginary unit $i$ defined as?
What is the imaginary unit $i$ defined as?
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$i=\sqrt{-1}$. Fundamental definition of the imaginary unit.
$i=\sqrt{-1}$. Fundamental definition of the imaginary unit.
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What is the factored form pattern for a monic quadratic $x^2+bx+c$?
What is the factored form pattern for a monic quadratic $x^2+bx+c$?
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$(x+m)(x+n)$ with $m+n=b$ and $mn=c$. Two numbers that add to $b$ and multiply to $c$.
$(x+m)(x+n)$ with $m+n=b$ and $mn=c$. Two numbers that add to $b$ and multiply to $c$.
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What are the solutions of $x^2=0$?
What are the solutions of $x^2=0$?
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$x=0$. Only zero squared equals zero.
$x=0$. Only zero squared equals zero.
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What is $\sqrt{-81}$ written using $i$?
What is $\sqrt{-81}$ written using $i$?
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$9i$. $\sqrt{-81}=\sqrt{81}\cdot i=9i$.
$9i$. $\sqrt{-81}=\sqrt{81}\cdot i=9i$.
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Solve by factoring: $3x^2-12x=0$.
Solve by factoring: $3x^2-12x=0$.
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$x=0$ or $x=4$. Factor out $3x$: $3x(x-4)=0$.
$x=0$ or $x=4$. Factor out $3x$: $3x(x-4)=0$.
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What does $\Delta=0$ tell you about the solutions of a quadratic?
What does $\Delta=0$ tell you about the solutions of a quadratic?
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One real double solution. Zero discriminant means the quadratic has one repeated root.
One real double solution. Zero discriminant means the quadratic has one repeated root.
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Solve by taking square roots: $(x-5)^2=0$.
Solve by taking square roots: $(x-5)^2=0$.
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$x=5$. Perfect square equals zero has one solution.
$x=5$. Perfect square equals zero has one solution.
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Solve by factoring: $x^2+5x+6=0$.
Solve by factoring: $x^2+5x+6=0$.
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$x=-3$ or $x=-2$. Find factors of 6 that add to 5: $(x+3)(x+2)=0$.
$x=-3$ or $x=-2$. Find factors of 6 that add to 5: $(x+3)(x+2)=0$.
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Solve by factoring: $x^2-7x+12=0$.
Solve by factoring: $x^2-7x+12=0$.
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$x=3$ or $x=4$. Find factors of 12 that add to $-7$: $(x-3)(x-4)=0$.
$x=3$ or $x=4$. Find factors of 12 that add to $-7$: $(x-3)(x-4)=0$.
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What does $\Delta>0$ tell you about the solutions of a quadratic?
What does $\Delta>0$ tell you about the solutions of a quadratic?
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Two distinct real solutions. Positive discriminant means two real roots exist.
Two distinct real solutions. Positive discriminant means two real roots exist.
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How do you rewrite $\sqrt{-k}$ for real $k>0$ using $i$?
How do you rewrite $\sqrt{-k}$ for real $k>0$ using $i$?
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$\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the square root.
$\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the square root.
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What is the key first step to solve $ax^2=k$ by taking square roots?
What is the key first step to solve $ax^2=k$ by taking square roots?
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Divide by $a$ to get $x^2=\frac{k}{a}$. Isolate $x^2$ to apply square root method.
Divide by $a$ to get $x^2=\frac{k}{a}$. Isolate $x^2$ to apply square root method.
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What are the solutions of $x^2=-m$ for real $m>0$?
What are the solutions of $x^2=-m$ for real $m>0$?
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$x=\pm i\sqrt{m}$. Negative under square root creates imaginary solutions.
$x=\pm i\sqrt{m}$. Negative under square root creates imaginary solutions.
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What is the vertex form used after completing the square?
What is the vertex form used after completing the square?
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$a(x-h)^2+k=0$ or $a(x-h)^2=-k$. Vertex form after completing the square process.
$a(x-h)^2+k=0$ or $a(x-h)^2=-k$. Vertex form after completing the square process.
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What number do you add and subtract to complete the square for $x^2+bx$?
What number do you add and subtract to complete the square for $x^2+bx$?
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$\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.
$\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.
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What property allows you to set each factor to $0$ in $(x-r)(x-s)=0$?
What property allows you to set each factor to $0$ in $(x-r)(x-s)=0$?
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Zero Product Property. If a product equals zero, at least one factor must be zero.
Zero Product Property. If a product equals zero, at least one factor must be zero.
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What are the solutions of $(x-r)(x-s)=0$?
What are the solutions of $(x-r)(x-s)=0$?
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$x=r$ or $x=s$. Set each factor equal to zero and solve.
$x=r$ or $x=s$. Set each factor equal to zero and solve.
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What is the square root of $a^2$ for real $a$?
What is the square root of $a^2$ for real $a$?
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$\sqrt{a^2}=|a|$. Principal square root is always non-negative.
$\sqrt{a^2}=|a|$. Principal square root is always non-negative.
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What are the solutions of $x^2=\frac{1}{4}$?
What are the solutions of $x^2=\frac{1}{4}$?
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$x=\pm\frac{1}{2}$. Perfect square: $\frac{1}{4}=\left(\frac{1}{2}\right)^2$.
$x=\pm\frac{1}{2}$. Perfect square: $\frac{1}{4}=\left(\frac{1}{2}\right)^2$.
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Solve by completing the square: $x^2-4x-1=0$.
Solve by completing the square: $x^2-4x-1=0$.
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$x=2\pm\sqrt{5}$. Complete square: $(x-2)^2=5$, so $x=2\pm\sqrt{5}$.
$x=2\pm\sqrt{5}$. Complete square: $(x-2)^2=5$, so $x=2\pm\sqrt{5}$.
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Solve using the quadratic formula: $x^2-2x-3=0$.
Solve using the quadratic formula: $x^2-2x-3=0$.
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$x=-1$ or $x=3$. Apply formula with $a=1$, $b=-2$, $c=-3$.
$x=-1$ or $x=3$. Apply formula with $a=1$, $b=-2$, $c=-3$.
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Solve by factoring: $x^2-9=0$.
Solve by factoring: $x^2-9=0$.
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$x=-3$ or $x=3$. Difference of squares: $(x-3)(x+3)=0$.
$x=-3$ or $x=3$. Difference of squares: $(x-3)(x+3)=0$.
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Solve by factoring: $4x^2-25=0$.
Solve by factoring: $4x^2-25=0$.
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$x=-\frac{5}{2}$ or $x=\frac{5}{2}$. Difference of squares: $(2x-5)(2x+5)=0$.
$x=-\frac{5}{2}$ or $x=\frac{5}{2}$. Difference of squares: $(2x-5)(2x+5)=0$.
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Solve by factoring: $x^2+2x-15=0$.
Solve by factoring: $x^2+2x-15=0$.
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$x=-5$ or $x=3$. Find factors of $-15$ that add to 2: $(x+5)(x-3)=0$.
$x=-5$ or $x=3$. Find factors of $-15$ that add to 2: $(x+5)(x-3)=0$.
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Solve by factoring: $x^2-4x+4=0$.
Solve by factoring: $x^2-4x+4=0$.
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$x=2$. Perfect square trinomial: $(x-2)^2=0$.
$x=2$. Perfect square trinomial: $(x-2)^2=0$.
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Solve by factoring: $2x^2+7x+3=0$.
Solve by factoring: $2x^2+7x+3=0$.
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$x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$.
$x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$.
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Solve by factoring: $3x^2-12x=0$.
Solve by factoring: $3x^2-12x=0$.
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$x=0$ or $x=4$. Factor out $3x$: $3x(x-4)=0$.
$x=0$ or $x=4$. Factor out $3x$: $3x(x-4)=0$.
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Solve using the quadratic formula: $2x^2+4x+1=0$.
Solve using the quadratic formula: $2x^2+4x+1=0$.
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$x=-1\pm\frac{\sqrt{2}}{2}$. Apply formula with $a=2$, $b=4$, $c=1$.
$x=-1\pm\frac{\sqrt{2}}{2}$. Apply formula with $a=2$, $b=4$, $c=1$.
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What is the discriminant of $x^2-6x+9=0$?
What is the discriminant of $x^2-6x+9=0$?
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$0$. $b^2-4ac=36-36=0$ for perfect square.
$0$. $b^2-4ac=36-36=0$ for perfect square.
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