Algebra Flashcards: Solve Quadratics By Multiple Methods

What this deck covers

This deck focuses on Solve Quadratics By Multiple Methods, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

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

Answer: $\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the square root.

Flashcard 2: Identify the solution type from $\Delta=0$.

Answer: One real double solution. Zero discriminant means one repeated solution.

Flashcard 3: Solve by factoring: $3x^2-12x=0$.

Answer: $x=0$ or $x=4$. Factor out $3x$: $3x(x-4)=0$.

Flashcard 4: What does $\Delta>0$ tell you about the solutions of a quadratic?

Answer: Two distinct real solutions. Positive discriminant means two real roots exist.

Flashcard 5: What is the discriminant of $x^2-6x+9=0$?

Answer: $0$. $b^2-4ac=36-36=0$ for perfect square.

Flashcard 6: What is the vertex form used after completing the square?

Answer: $a(x-h)^2+k=0$ or $a(x-h)^2=-k$. Vertex form after completing the square process.

Flashcard 7: How do you rewrite $\sqrt{-k}$ for real $k>0$ using $i$?

Answer: $\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the square root.

Flashcard 8: Identify the solution type from $\Delta=25$.

Answer: Two distinct real solutions. Positive discriminant means two real solutions.

Flashcard 9: What is the quadratic formula for solutions of $ax^2+bx+c=0$?

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Standard formula derived from completing the square on $ax^2+bx+c=0$.

Flashcard 10: What does $\Delta=0$ tell you about the solutions of a quadratic?

Answer: One real double solution. Zero discriminant means the quadratic has one repeated root.

Flashcard 11: What is the square root of $a^2$ for real $a$?

Answer: $\sqrt{a^2}=|a|$. Principal square root is always non-negative.

Flashcard 12: What are the solutions of $x^2=0$?

Answer: $x=0$. Only zero squared equals zero.

Flashcard 13: What number do you add and subtract to complete the square for $x^2+bx$?

Answer: $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.

Flashcard 14: What are the solutions of $(x-r)(x-s)=0$?

Answer: $x=r$ or $x=s$. Set each factor equal to zero and solve.

Flashcard 15: What does $\Delta<0$ tell you about the solutions of a quadratic?

Answer: Two complex (nonreal) solutions. Negative discriminant means roots involve imaginary numbers.

Flashcard 16: What is the imaginary unit $i$ defined as?

Answer: $i=\sqrt{-1}$. Fundamental definition of the imaginary unit.

Flashcard 17: Solve by factoring: $x^2+5x+6=0$.

Answer: $x=-3$ or $x=-2$. Find factors of 6 that add to 5: $(x+3)(x+2)=0$.

Flashcard 18: Solve by taking square roots: $(x-5)^2=0$.

Answer: $x=5$. Perfect square equals zero has one solution.

Flashcard 19: Solve using the quadratic formula: $2x^2+4x+1=0$.

Answer: $x=-1\pm\frac{\sqrt{2}}{2}$. Apply formula with $a=2$, $b=4$, $c=1$.

Flashcard 20: What is the key first step to solve $ax^2=k$ by taking square roots?

Answer: Divide by $a$ to get $x^2=\frac{k}{a}$. Isolate $x^2$ to apply square root method.

Flashcard 21: Solve by factoring: $4x^2-25=0$.

Answer: $x=-\frac{5}{2}$ or $x=\frac{5}{2}$. Difference of squares: $(2x-5)(2x+5)=0$.

Flashcard 22: What does $\Delta>0$ tell you about the solutions of a quadratic?

Answer: Two distinct real solutions. Positive discriminant means two real roots exist.

Flashcard 23: Solve by factoring: $x^2-4x+4=0$.

Answer: $x=2$. Perfect square trinomial: $(x-2)^2=0$.

Flashcard 24: Solve by factoring: $x^2-7x+12=0$.

Answer: $x=3$ or $x=4$. Find factors of 12 that add to $-7$: $(x-3)(x-4)=0$.

Flashcard 25: What is the factored form pattern for a monic quadratic $x^2+bx+c$?

Answer: $(x+m)(x+n)$ with $m+n=b$ and $mn=c$. Two numbers that add to $b$ and multiply to $c$.

Flashcard 26: Identify the solution type from $\Delta=0$.

Answer: One real double solution. Zero discriminant means one repeated solution.

Flashcard 27: Solve using the quadratic formula: $x^2-2x-3=0$.

Answer: $x=-1$ or $x=3$. Apply formula with $a=1$, $b=-2$, $c=-3$.

Flashcard 28: Solve by completing the square: $x^2-4x-1=0$.

Answer: $x=2\pm\sqrt{5}$. Complete square: $(x-2)^2=5$, so $x=2\pm\sqrt{5}$.

Flashcard 29: What is the factored form pattern for a monic quadratic $x^2+bx+c$?

Answer: $(x+m)(x+n)$ with $m+n=b$ and $mn=c$. Two numbers that add to $b$ and multiply to $c$.

Flashcard 30: Solve using the quadratic formula: $2x^2+4x+1=0$.

Answer: $x=-1\pm\frac{\sqrt{2}}{2}$. Apply formula with $a=2$, $b=4$, $c=1$.

Flashcard 31: What property allows you to set each factor to $0$ in $(x-r)(x-s)=0$?

Answer: Zero Product Property. If a product equals zero, at least one factor must be zero.

Flashcard 32: What is $\sqrt{\frac{49}{16}}$?

Answer: $\frac{7}{4}$. Square root of fraction: $\sqrt{\frac{49}{16}}=\frac{7}{4}$.

Flashcard 33: What is $\sqrt{36}$?

Answer: $6$. Perfect square: $36=6^2$.

Flashcard 34: What is $\sqrt{-81}$ written using $i$?

Answer: $9i$. $\sqrt{-81}=\sqrt{81}\cdot i=9i$.

Flashcard 35: Identify the solution type from $\Delta=25$.

Answer: Two distinct real solutions. Positive discriminant means two real solutions.

Flashcard 36: What does $\Delta<0$ tell you about the solutions of a quadratic?

Answer: Two complex (nonreal) solutions. Negative discriminant means roots involve imaginary numbers.

Flashcard 37: Solve by completing the square: $x^2-4x-1=0$.

Answer: $x=2\pm\sqrt{5}$. Complete square: $(x-2)^2=5$, so $x=2\pm\sqrt{5}$.

Flashcard 38: What is the discriminant of $x^2-6x+9=0$?

Answer: $0$. $b^2-4ac=36-36=0$ for perfect square.

Flashcard 39: What number do you add and subtract to complete the square for $x^2+bx$?

Answer: $\left(\frac{b}{2}\right)^2$. Half the coefficient of $x$, then squared.

Flashcard 40: What are the solutions of $x^2=-m$ for real $m>0$?

Answer: $x=\pm i\sqrt{m}$. Negative under square root creates imaginary solutions.

Flashcard 41: What is the key first step to solve $ax^2=k$ by taking square roots?

Answer: Divide by $a$ to get $x^2=\frac{k}{a}$. Isolate $x^2$ to apply square root method.

Flashcard 42: What is the discriminant of $x^2+2x+10=0$?

Answer: $-36$. $b^2-4ac=4-40=-36$ gives complex solutions.

Flashcard 43: Identify the solution type from $\Delta=-9$.

Answer: Two complex (nonreal) solutions. Negative discriminant means complex solutions.

Flashcard 44: What is the vertex form used after completing the square?

Answer: $a(x-h)^2+k=0$ or $a(x-h)^2=-k$. Vertex form after completing the square process.

Flashcard 45: What are the solutions of $x^2=-m$ for real $m>0$?

Answer: $x=\pm i\sqrt{m}$. Negative under square root creates imaginary solutions.

Flashcard 46: Solve by factoring: $3x^2-12x=0$.

Answer: $x=0$ or $x=4$. Factor out $3x$: $3x(x-4)=0$.

Flashcard 47: What are the solutions of $x^2=\frac{1}{4}$?

Answer: $x=\pm\frac{1}{2}$. Perfect square: $\frac{1}{4}=\left(\frac{1}{2}\right)^2$.

Flashcard 48: What is the discriminant of $x^2+2x+10=0$?

Answer: $-36$. $b^2-4ac=4-40=-36$ gives complex solutions.

Flashcard 49: What are the solutions of $(x-r)(x-s)=0$?

Answer: $x=r$ or $x=s$. Set each factor equal to zero and solve.

Flashcard 50: Solve by factoring: $4x^2-25=0$.

Answer: $x=-\frac{5}{2}$ or $x=\frac{5}{2}$. Difference of squares: $(2x-5)(2x+5)=0$.

Flashcard 51: Solve by factoring: $x^2+2x-15=0$.

Answer: $x=-5$ or $x=3$. Find factors of $-15$ that add to 2: $(x+5)(x-3)=0$.

Flashcard 52: Solve by factoring: $x^2-7x+12=0$.

Answer: $x=3$ or $x=4$. Find factors of 12 that add to $-7$: $(x-3)(x-4)=0$.

Flashcard 53: What is the square root of $a^2$ for real $a$?

Answer: $\sqrt{a^2}=|a|$. Principal square root is always non-negative.

Flashcard 54: Solve by factoring: $2x^2+7x+3=0$.

Answer: $x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$.

Flashcard 55: What is $\sqrt{-81}$ written using $i$?

Answer: $9i$. $\sqrt{-81}=\sqrt{81}\cdot i=9i$.

Flashcard 56: Solve by factoring: $x^2+5x+6=0$.

Answer: $x=-3$ or $x=-2$. Find factors of 6 that add to 5: $(x+3)(x+2)=0$.

Flashcard 57: Solve by taking square roots: $(x-5)^2=0$.

Answer: $x=5$. Perfect square equals zero has one solution.

Flashcard 58: What is $\sqrt{\frac{49}{16}}$?

Answer: $\frac{7}{4}$. Square root of fraction: $\sqrt{\frac{49}{16}}=\frac{7}{4}$.

Flashcard 59: What is the discriminant for $ax^2+bx+c=0$?

Answer: $\Delta=b^2-4ac$. Expression under the square root that determines solution types.

Flashcard 60: What are the solutions of $x^2=0$?

Answer: $x=0$. Only zero squared equals zero.

Flashcard 61: What is $\sqrt{36}$?

Answer: $6$. Perfect square: $36=6^2$.

Flashcard 62: Identify the solution type from $\Delta=-9$.

Answer: Two complex (nonreal) solutions. Negative discriminant means complex solutions.

Flashcard 63: What is the imaginary unit $i$ defined as?

Answer: $i=\sqrt{-1}$. Fundamental definition of the imaginary unit.

Flashcard 64: Solve using the quadratic formula: $x^2-2x-3=0$.

Answer: $x=-1$ or $x=3$. Apply formula with $a=1$, $b=-2$, $c=-3$.

Flashcard 65: What does $\Delta=0$ tell you about the solutions of a quadratic?

Answer: One real double solution. Zero discriminant means the quadratic has one repeated root.

Flashcard 66: Solve by factoring: $x^2+2x-15=0$.

Answer: $x=-5$ or $x=3$. Find factors of $-15$ that add to 2: $(x+5)(x-3)=0$.

Flashcard 67: Solve by factoring: $2x^2+7x+3=0$.

Answer: $x=-3$ or $x=-\frac{1}{2}$. Factor as $(2x+1)(x+3)=0$.

Flashcard 68: Solve by factoring: $x^2-9=0$.

Answer: $x=-3$ or $x=3$. Difference of squares: $(x-3)(x+3)=0$.