Algebra Flashcards: Solving Quadratic Equations With Complex Solutions

What this deck covers

This deck focuses on Solving Quadratic Equations With Complex Solutions, 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: State the vertex $x$-coordinate formula for $y=ax^2+bx+c$.

Answer: $x=-\frac{b}{2a}$. Found by setting the derivative equal to zero or completing the square.

Flashcard 2: Solve $5x^2+10x+13=0$.

Answer: $x=-1\pm \frac{2}{5}i$. $\Delta = 100 - 260 = -160 < 0$, then divide by $2a = 10$.

Flashcard 3: Simplify $\sqrt{-75}$ in simplest form.

Answer: $5\sqrt{3}i$. $\sqrt{-75} = \sqrt{25 \cdot 3} \cdot i = 5\sqrt{3}i$.

Flashcard 4: Simplify $\sqrt{-12}$ in simplest form.

Answer: $2\sqrt{3}i$. $\sqrt{-12} = \sqrt{4 \cdot 3} \cdot i = 2\sqrt{3}i$.

Flashcard 5: What is the standard simplification for $\sqrt{-k}$ when $k>0$?

Answer: $\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the radical: $\sqrt{-k} = \sqrt{(-1) \cdot k}$.

Flashcard 6: What is $\sqrt{-1}$ written as a complex unit?

Answer: $i$. The imaginary unit, defined as $i^2 = -1$.

Flashcard 7: Which condition on $\Delta$ guarantees two nonreal complex solutions?

Answer: $\Delta<0$. Negative discriminant means the square root involves $\sqrt{\text{negative}}$.

Flashcard 8: Identify whether $x^2+6x+9=0$ has real solutions or complex solutions.

Answer: Real solutions (a repeated real root). $\Delta = 36 - 36 = 0$, indicating one repeated real root.

Flashcard 9: Identify the complex conjugate of $a+bi$.

Answer: $a-bi$. Change the sign of the imaginary part.

Flashcard 10: Solve $(x+2)^2=-9$.

Answer: $x=-2\pm 3i$. Take square root: $x + 2 = \pm\sqrt{-9} = \pm 3i$.

Flashcard 11: Solve $x^2-6x+13=0$.

Answer: $x=3\pm 2i$. $\Delta = 36 - 52 = -16 < 0$, so solutions involve $i$.

Flashcard 12: Solve $4x^2+4x+5=0$.

Answer: $x=-\frac{1}{2}\pm i$. $\Delta = 16 - 80 = -64 < 0$, then divide by $2a = 8$.

Flashcard 13: Solve $x^2+18x+85=0$.

Answer: $x=-9\pm 2i$. $\Delta = 324 - 340 = -16 < 0$, giving complex solutions.

Flashcard 14: Solve $x^2+6x+34=0$.

Answer: $x=-3\pm 5i$. $\Delta = 36 - 136 = -100 < 0$, yielding complex solutions.

Flashcard 15: Solve $(x-3)^2=-16$.

Answer: $x=3\pm 4i$. Take square root: $x - 3 = \pm\sqrt{-16} = \pm 4i$.

Flashcard 16: Solve $x^2-14x+58=0$.

Answer: $x=7\pm 3i$. $\Delta = 196 - 232 = -36 < 0$, so solutions involve $i$.

Flashcard 17: Solve $x^2-18x+90=0$.

Answer: $x=9\pm 3i$. $\Delta = 324 - 360 = -36 < 0$, so solutions are complex.

Flashcard 18: Solve $x^2+16=0$.

Answer: $x=\pm 4i$. Rearrange to $x^2 = -16$, then $x = \pm\sqrt{-16} = \pm 4i$.

Flashcard 19: Solve $x^2+16=0$.

Answer: $x=\pm 4i$. Rearrange to $x^2 = -16$, then $x = \pm\sqrt{-16} = \pm 4i$.

Flashcard 20: What relationship do nonreal solutions have for a quadratic with real coefficients?

Answer: They occur as conjugate pairs $a\pm bi$. Complex Conjugate Root Theorem for polynomials with real coefficients.

Flashcard 21: Solve $x^2+4x+5=0$.

Answer: $x=-2\pm i$. $\Delta = 16 - 20 = -4 < 0$, so use quadratic formula with $i$.

Flashcard 22: Solve $x^2+14x+53=0$.

Answer: $x=-7\pm 2i$. $\Delta = 196 - 212 = -16 < 0$, yielding complex solutions.

Flashcard 23: Solve $x^2+6x+34=0$.

Answer: $x=-3\pm 5i$. $\Delta = 36 - 136 = -100 < 0$, yielding complex solutions.

Flashcard 24: What is the discriminant of $ax^2+bx+c=0$?

Answer: $\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.

Flashcard 25: Solve $x^2= -49$.

Answer: $x=\pm 7i$. Take square root of both sides: $x = \pm\sqrt{-49} = \pm 7i$.

Flashcard 26: Identify the sum of roots of $ax^2+bx+c=0$ in terms of $a$ and $b$.

Answer: $r_1+r_2=-\frac{b}{a}$. Vieta's formula: sum of roots equals $-\frac{\text{coefficient of } x}{\text{leading coefficient}}$.

Flashcard 27: Find the discriminant of $x^2+4x+5=0$.

Answer: $\Delta=-4$. $\Delta = b^2 - 4ac = 16 - 20 = -4$.

Flashcard 28: Identify the complex conjugate of $a+bi$.

Answer: $a-bi$. Change the sign of the imaginary part.

Flashcard 29: Solve $x^2-12x+52=0$.

Answer: $x=6\pm 4i$. $\Delta = 144 - 208 = -64 < 0$, so solutions are complex.

Flashcard 30: State the result of completing the square on $x^2+px$.

Answer: $x^2+px=\left(x+\frac{p}{2}\right)^2-\left(\frac{p}{2}\right)^2$. Add and subtract $(\frac{p}{2})^2$ to create a perfect square trinomial.

Flashcard 31: Solve $x^2-2x+2=0$.

Answer: $x=1\pm i$. $\Delta = 4 - 8 = -4 < 0$, giving complex solutions.

Flashcard 32: What is the sum of the solutions to $x^2+4x+5=0$?

Answer: $-4$. Sum of roots $= -\frac{b}{a} = -\frac{4}{1} = -4$.

Flashcard 33: State the vertex $x$-coordinate formula for $y=ax^2+bx+c$.

Answer: $x=-\frac{b}{2a}$. Found by setting the derivative equal to zero or completing the square.

Flashcard 34: Which condition on $\Delta$ guarantees two nonreal complex solutions?

Answer: $\Delta<0$. Negative discriminant means the square root involves $\sqrt{\text{negative}}$.

Flashcard 35: What are the solutions of $x^2+1=0$?

Answer: $x=\pm i$. Rearrange to $x^2 = -1$, then $x = \pm\sqrt{-1} = \pm i$.

Flashcard 36: What is $\sqrt{-1}$ written as a complex unit?

Answer: $i$. The imaginary unit, defined as $i^2 = -1$.

Flashcard 37: What is the discriminant of $ax^2+bx+c=0$?

Answer: $\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.

Flashcard 38: Solve $x^2-4x+8=0$.

Answer: $x=2\pm 2i$. $\Delta = 16 - 32 = -16 < 0$, so solutions are complex.

Flashcard 39: Solve $x^2+4x+5=0$.

Answer: $x=-2\pm i$. $\Delta = 16 - 20 = -4 < 0$, so use quadratic formula with $i$.

Flashcard 40: Solve $x^2-6x+13=0$.

Answer: $x=3\pm 2i$. $\Delta = 36 - 52 = -16 < 0$, so solutions involve $i$.

Flashcard 41: Identify the real part and imaginary part of the solution $3-2i$.

Answer: Real part $3$, imaginary part $-2$. In $a + bi$ form, $a$ is real part and $b$ is imaginary part.

Flashcard 42: Find the discriminant of $3x^2-6x+5=0$.

Answer: $\Delta=-24$. $\Delta = b^2 - 4ac = 36 - 60 = -24$.

Flashcard 43: What are the solutions of $x^2+1=0$?

Answer: $x=\pm i$. Rearrange to $x^2 = -1$, then $x = \pm\sqrt{-1} = \pm i$.

Flashcard 44: Solve $x^2-8x+20=0$ by completing the square.

Answer: $x=4\pm 2i$. Complete the square: $(x - 4)^2 = -4$, so $x - 4 = \pm 2i$.

Flashcard 45: State the result of completing the square on $x^2+px$.

Answer: $x^2+px=\left(x+\frac{p}{2}\right)^2-\left(\frac{p}{2}\right)^2$. Add and subtract $(\frac{p}{2})^2$ to create a perfect square trinomial.

Flashcard 46: Solve $2x^2-8x+17=0$.

Answer: $x=2\pm \frac{\sqrt{2}}{2}i$. $\Delta = 64 - 136 = -72 < 0$, then divide by $2a = 4$.

Flashcard 47: Solve $x^2+18x+85=0$.

Answer: $x=-9\pm 2i$. $\Delta = 324 - 340 = -16 < 0$, giving complex solutions.

Flashcard 48: What is the product of the solutions to $x^2+4x+5=0$?

Answer: $5$. Product of roots $= \frac{c}{a} = \frac{5}{1} = 5$.

Flashcard 49: Solve $x^2+2x+10=0$.

Answer: $x=-1\pm 3i$. $\Delta = 4 - 40 = -36 < 0$, so solutions are complex.

Flashcard 50: Identify the sum of roots of $ax^2+bx+c=0$ in terms of $a$ and $b$.

Answer: $r_1+r_2=-\frac{b}{a}$. Vieta's formula: sum of roots equals $-\frac{\text{coefficient of } x}{\text{leading coefficient}}$.

Flashcard 51: Solve $x^2+2x+5=0$ by completing the square.

Answer: $x=-1\pm 2i$. Complete the square: $(x + 1)^2 = -4$, so $x + 1 = \pm 2i$.

Flashcard 52: What relationship do nonreal solutions have for a quadratic with real coefficients?

Answer: They occur as conjugate pairs $a\pm bi$. Complex Conjugate Root Theorem for polynomials with real coefficients.

Flashcard 53: Solve $(x+2)^2=-9$.

Answer: $x=-2\pm 3i$. Take square root: $x + 2 = \pm\sqrt{-9} = \pm 3i$.

Flashcard 54: Solve $x^2-12x+52=0$.

Answer: $x=6\pm 4i$. $\Delta = 144 - 208 = -64 < 0$, so solutions are complex.

Flashcard 55: Simplify $\sqrt{-36}$.

Answer: $6i$. $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$.

Flashcard 56: Solve $x^2-10x+29=0$.

Answer: $x=5\pm 2i$. $\Delta = 100 - 116 = -16 < 0$, so solutions involve $i$.

Flashcard 57: Solve $x^2+2x+10=0$.

Answer: $x=-1\pm 3i$. $\Delta = 4 - 40 = -36 < 0$, so solutions are complex.

Flashcard 58: Solve $x^2+8x+20=0$.

Answer: $x=-4\pm 2i$. $\Delta = 64 - 80 = -16 < 0$, so solutions are nonreal.

Flashcard 59: Solve $x^2-14x+58=0$.

Answer: $x=7\pm 3i$. $\Delta = 196 - 232 = -36 < 0$, so solutions involve $i$.

Flashcard 60: Solve $(2x-1)^2=-25$.

Answer: $x=\frac{1}{2}\pm \frac{5}{2}i$. Take square root: $2x - 1 = \pm\sqrt{-25} = \pm 5i$.

Flashcard 61: Solve $x^2-18x+90=0$.

Answer: $x=9\pm 3i$. $\Delta = 324 - 360 = -36 < 0$, so solutions are complex.

Flashcard 62: Solve $3x^2+6x+7=0$.

Answer: $x=-1\pm \frac{\sqrt{3}}{3}i$. $\Delta = 36 - 84 = -48 < 0$, then divide by $2a = 6$.

Flashcard 63: Solve $(x-3)^2=-16$.

Answer: $x=3\pm 4i$. Take square root: $x - 3 = \pm\sqrt{-16} = \pm 4i$.

Flashcard 64: What is the standard simplification for $\sqrt{-k}$ when $k>0$?

Answer: $\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the radical: $\sqrt{-k} = \sqrt{(-1) \cdot k}$.

Flashcard 65: State the quadratic formula for solutions of $ax^2+bx+c=0$.

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square or using algebraic manipulation.

Flashcard 66: Solve $x^2-4x+8=0$.

Answer: $x=2\pm 2i$. $\Delta = 16 - 32 = -16 < 0$, so solutions are complex.

Flashcard 67: Solve $5x^2+10x+13=0$.

Answer: $x=-1\pm \frac{2}{5}i$. $\Delta = 100 - 260 = -160 < 0$, then divide by $2a = 10$.

Flashcard 68: Solve $x^2-2x+2=0$.

Answer: $x=1\pm i$. $\Delta = 4 - 8 = -4 < 0$, giving complex solutions.

Flashcard 69: Simplify $\sqrt{-36}$.

Answer: $6i$. $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$.

Flashcard 70: Solve $x^2+14x+53=0$.

Answer: $x=-7\pm 2i$. $\Delta = 196 - 212 = -16 < 0$, yielding complex solutions.

Flashcard 71: Solve $3x^2+6x+7=0$.

Answer: $x=-1\pm \frac{\sqrt{3}}{3}i$. $\Delta = 36 - 84 = -48 < 0$, then divide by $2a = 6$.

Flashcard 72: What are the solutions of $x^2+9=0$?

Answer: $x=\pm 3i$. Rearrange to $x^2 = -9$, then $x = \pm\sqrt{-9} = \pm 3i$.

Flashcard 73: What are the solutions of $x^2+9=0$?

Answer: $x=\pm 3i$. Rearrange to $x^2 = -9$, then $x = \pm\sqrt{-9} = \pm 3i$.

Flashcard 74: Solve $x^2+8x+20=0$.

Answer: $x=-4\pm 2i$. $\Delta = 64 - 80 = -16 < 0$, so solutions are nonreal.

Flashcard 75: Simplify $\sqrt{-8}$ in simplest form.

Answer: $2\sqrt{2}i$. $\sqrt{-8} = \sqrt{4 \cdot 2} \cdot i = 2\sqrt{2}i$.

Flashcard 76: Identify the real part and imaginary part of the solution $3-2i$.

Answer: Real part $3$, imaginary part $-2$. In $a + bi$ form, $a$ is real part and $b$ is imaginary part.

Flashcard 77: Identify whether $x^2-2x+5=0$ has real solutions or complex solutions.

Answer: Complex solutions (nonreal). $\Delta = 4 - 20 = -16 < 0$, indicating nonreal solutions.

Flashcard 78: Identify whether $x^2-2x+5=0$ has real solutions or complex solutions.

Answer: Complex solutions (nonreal). $\Delta = 4 - 20 = -16 < 0$, indicating nonreal solutions.

Flashcard 79: Identify the product of roots of $ax^2+bx+c=0$ in terms of $a$ and $c$.

Answer: $r_1r_2=\frac{c}{a}$. Vieta's formula: product of roots equals $\frac{\text{constant term}}{\text{leading coefficient}}$.

Flashcard 80: Solve $x^2+12x+40=0$.

Answer: $x=-6\pm 2i$. $\Delta = 144 - 160 = -16 < 0$, giving complex solutions.

Flashcard 81: State the quadratic formula for solutions of $ax^2+bx+c=0$.

Answer: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square or using algebraic manipulation.

Flashcard 82: Solve $2x^2+4x+5=0$.

Answer: $x=-1\pm \frac{\sqrt{6}}{2}i$. $\Delta = 16 - 40 = -24 < 0$, then divide by $2a = 4$.

Flashcard 83: Solve $x^2+10x+26=0$.

Answer: $x=-5\pm i$. $\Delta = 100 - 104 = -4 < 0$, yielding complex solutions.