Solving Quadratic Equations with Complex Solutions - Algebra
Card 1 of 30
What is the discriminant of $ax^2+bx+c=0$?
What is the discriminant of $ax^2+bx+c=0$?
Tap to reveal answer
$\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.
$\Delta=b^2-4ac$. The expression under the square root in the quadratic formula.
← Didn't Know|Knew It →
Which condition on $\Delta$ guarantees two nonreal complex solutions?
Which condition on $\Delta$ guarantees two nonreal complex solutions?
Tap to reveal answer
$\Delta<0$. Negative discriminant means the square root involves $\sqrt{\text{negative}}$.
$\Delta<0$. Negative discriminant means the square root involves $\sqrt{\text{negative}}$.
← Didn't Know|Knew It →
What is $\sqrt{-1}$ written as a complex unit?
What is $\sqrt{-1}$ written as a complex unit?
Tap to reveal answer
$i$. The imaginary unit, defined as $i^2 = -1$.
$i$. The imaginary unit, defined as $i^2 = -1$.
← Didn't Know|Knew It →
Solve $x^2-10x+29=0$.
Solve $x^2-10x+29=0$.
Tap to reveal answer
$x=5\pm 2i$. $\Delta = 100 - 116 = -16 < 0$, so solutions involve $i$.
$x=5\pm 2i$. $\Delta = 100 - 116 = -16 < 0$, so solutions involve $i$.
← Didn't Know|Knew It →
Solve $4x^2+4x+5=0$.
Solve $4x^2+4x+5=0$.
Tap to reveal answer
$x=-\frac{1}{2}\pm i$. $\Delta = 16 - 80 = -64 < 0$, then divide by $2a = 8$.
$x=-\frac{1}{2}\pm i$. $\Delta = 16 - 80 = -64 < 0$, then divide by $2a = 8$.
← Didn't Know|Knew It →
Solve $x^2+2x+5=0$ by completing the square.
Solve $x^2+2x+5=0$ by completing the square.
Tap to reveal answer
$x=-1\pm 2i$. Complete the square: $(x + 1)^2 = -4$, so $x + 1 = \pm 2i$.
$x=-1\pm 2i$. Complete the square: $(x + 1)^2 = -4$, so $x + 1 = \pm 2i$.
← Didn't Know|Knew It →
Solve $(x+2)^2=-9$.
Solve $(x+2)^2=-9$.
Tap to reveal answer
$x=-2\pm 3i$. Take square root: $x + 2 = \pm\sqrt{-9} = \pm 3i$.
$x=-2\pm 3i$. Take square root: $x + 2 = \pm\sqrt{-9} = \pm 3i$.
← Didn't Know|Knew It →
Solve $(x-3)^2=-16$.
Solve $(x-3)^2=-16$.
Tap to reveal answer
$x=3\pm 4i$. Take square root: $x - 3 = \pm\sqrt{-16} = \pm 4i$.
$x=3\pm 4i$. Take square root: $x - 3 = \pm\sqrt{-16} = \pm 4i$.
← Didn't Know|Knew It →
Solve $(2x-1)^2=-25$.
Solve $(2x-1)^2=-25$.
Tap to reveal answer
$x=\frac{1}{2}\pm \frac{5}{2}i$. Take square root: $2x - 1 = \pm\sqrt{-25} = \pm 5i$.
$x=\frac{1}{2}\pm \frac{5}{2}i$. Take square root: $2x - 1 = \pm\sqrt{-25} = \pm 5i$.
← Didn't Know|Knew It →
Simplify $\sqrt{-8}$ in simplest form.
Simplify $\sqrt{-8}$ in simplest form.
Tap to reveal answer
$2\sqrt{2}i$. $\sqrt{-8} = \sqrt{4 \cdot 2} \cdot i = 2\sqrt{2}i$.
$2\sqrt{2}i$. $\sqrt{-8} = \sqrt{4 \cdot 2} \cdot i = 2\sqrt{2}i$.
← Didn't Know|Knew It →
What are the solutions of $x^2+1=0$?
What are the solutions of $x^2+1=0$?
Tap to reveal answer
$x=\pm i$. Rearrange to $x^2 = -1$, then $x = \pm\sqrt{-1} = \pm i$.
$x=\pm i$. Rearrange to $x^2 = -1$, then $x = \pm\sqrt{-1} = \pm i$.
← Didn't Know|Knew It →
State the quadratic formula for solutions of $ax^2+bx+c=0$.
State the quadratic formula for solutions of $ax^2+bx+c=0$.
Tap to reveal answer
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square or using algebraic manipulation.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived by completing the square or using algebraic manipulation.
← Didn't Know|Knew It →
What is the standard simplification for $\sqrt{-k}$ when $k>0$?
What is the standard simplification for $\sqrt{-k}$ when $k>0$?
Tap to reveal answer
$\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the radical: $\sqrt{-k} = \sqrt{(-1) \cdot k}$.
$\sqrt{-k}=i\sqrt{k}$. Factor out $-1$ from under the radical: $\sqrt{-k} = \sqrt{(-1) \cdot k}$.
← Didn't Know|Knew It →
Identify the complex conjugate of $a+bi$.
Identify the complex conjugate of $a+bi$.
Tap to reveal answer
$a-bi$. Change the sign of the imaginary part.
$a-bi$. Change the sign of the imaginary part.
← Didn't Know|Knew It →
What relationship do nonreal solutions have for a quadratic with real coefficients?
What relationship do nonreal solutions have for a quadratic with real coefficients?
Tap to reveal answer
They occur as conjugate pairs $a\pm bi$. Complex Conjugate Root Theorem for polynomials with real coefficients.
They occur as conjugate pairs $a\pm bi$. Complex Conjugate Root Theorem for polynomials with real coefficients.
← Didn't Know|Knew It →
State the vertex $x$-coordinate formula for $y=ax^2+bx+c$.
State the vertex $x$-coordinate formula for $y=ax^2+bx+c$.
Tap to reveal answer
$x=-\frac{b}{2a}$. Found by setting the derivative equal to zero or completing the square.
$x=-\frac{b}{2a}$. Found by setting the derivative equal to zero or completing the square.
← Didn't Know|Knew It →
State the result of completing the square on $x^2+px$.
State the result of completing the square on $x^2+px$.
Tap to reveal 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.
$x^2+px=\left(x+\frac{p}{2}\right)^2-\left(\frac{p}{2}\right)^2$. Add and subtract $(\frac{p}{2})^2$ to create a perfect square trinomial.
← Didn't Know|Knew It →
Solve $x^2+4x+5=0$.
Solve $x^2+4x+5=0$.
Tap to reveal answer
$x=-2\pm i$. $\Delta = 16 - 20 = -4 < 0$, so use quadratic formula with $i$.
$x=-2\pm i$. $\Delta = 16 - 20 = -4 < 0$, so use quadratic formula with $i$.
← Didn't Know|Knew It →
Solve $x^2-6x+13=0$.
Solve $x^2-6x+13=0$.
Tap to reveal answer
$x=3\pm 2i$. $\Delta = 36 - 52 = -16 < 0$, so solutions involve $i$.
$x=3\pm 2i$. $\Delta = 36 - 52 = -16 < 0$, so solutions involve $i$.
← Didn't Know|Knew It →
Solve $x^2+2x+10=0$.
Solve $x^2+2x+10=0$.
Tap to reveal answer
$x=-1\pm 3i$. $\Delta = 4 - 40 = -36 < 0$, so solutions are complex.
$x=-1\pm 3i$. $\Delta = 4 - 40 = -36 < 0$, so solutions are complex.
← Didn't Know|Knew It →
Solve $x^2-2x+2=0$.
Solve $x^2-2x+2=0$.
Tap to reveal answer
$x=1\pm i$. $\Delta = 4 - 8 = -4 < 0$, giving complex solutions.
$x=1\pm i$. $\Delta = 4 - 8 = -4 < 0$, giving complex solutions.
← Didn't Know|Knew It →
Solve $x^2+8x+20=0$.
Solve $x^2+8x+20=0$.
Tap to reveal answer
$x=-4\pm 2i$. $\Delta = 64 - 80 = -16 < 0$, so solutions are nonreal.
$x=-4\pm 2i$. $\Delta = 64 - 80 = -16 < 0$, so solutions are nonreal.
← Didn't Know|Knew It →
Solve $x^2+6x+34=0$.
Solve $x^2+6x+34=0$.
Tap to reveal answer
$x=-3\pm 5i$. $\Delta = 36 - 136 = -100 < 0$, yielding complex solutions.
$x=-3\pm 5i$. $\Delta = 36 - 136 = -100 < 0$, yielding complex solutions.
← Didn't Know|Knew It →
Solve $x^2+12x+40=0$.
Solve $x^2+12x+40=0$.
Tap to reveal answer
$x=-6\pm 2i$. $\Delta = 144 - 160 = -16 < 0$, giving complex solutions.
$x=-6\pm 2i$. $\Delta = 144 - 160 = -16 < 0$, giving complex solutions.
← Didn't Know|Knew It →
Solve $x^2-4x+8=0$.
Solve $x^2-4x+8=0$.
Tap to reveal answer
$x=2\pm 2i$. $\Delta = 16 - 32 = -16 < 0$, so solutions are complex.
$x=2\pm 2i$. $\Delta = 16 - 32 = -16 < 0$, so solutions are complex.
← Didn't Know|Knew It →
Solve $x^2+10x+26=0$.
Solve $x^2+10x+26=0$.
Tap to reveal answer
$x=-5\pm i$. $\Delta = 100 - 104 = -4 < 0$, yielding complex solutions.
$x=-5\pm i$. $\Delta = 100 - 104 = -4 < 0$, yielding complex solutions.
← Didn't Know|Knew It →
Solve $2x^2+4x+5=0$.
Solve $2x^2+4x+5=0$.
Tap to reveal answer
$x=-1\pm \frac{\sqrt{6}}{2}i$. $\Delta = 16 - 40 = -24 < 0$, then divide by $2a = 4$.
$x=-1\pm \frac{\sqrt{6}}{2}i$. $\Delta = 16 - 40 = -24 < 0$, then divide by $2a = 4$.
← Didn't Know|Knew It →
Solve $3x^2+6x+7=0$.
Solve $3x^2+6x+7=0$.
Tap to reveal answer
$x=-1\pm \frac{\sqrt{3}}{3}i$. $\Delta = 36 - 84 = -48 < 0$, then divide by $2a = 6$.
$x=-1\pm \frac{\sqrt{3}}{3}i$. $\Delta = 36 - 84 = -48 < 0$, then divide by $2a = 6$.
← Didn't Know|Knew It →
Solve $2x^2-8x+17=0$.
Solve $2x^2-8x+17=0$.
Tap to reveal answer
$x=2\pm \frac{\sqrt{2}}{2}i$. $\Delta = 64 - 136 = -72 < 0$, then divide by $2a = 4$.
$x=2\pm \frac{\sqrt{2}}{2}i$. $\Delta = 64 - 136 = -72 < 0$, then divide by $2a = 4$.
← Didn't Know|Knew It →
Solve $5x^2+10x+13=0$.
Solve $5x^2+10x+13=0$.
Tap to reveal answer
$x=-1\pm \frac{2}{5}i$. $\Delta = 100 - 260 = -160 < 0$, then divide by $2a = 10$.
$x=-1\pm \frac{2}{5}i$. $\Delta = 100 - 260 = -160 < 0$, then divide by $2a = 10$.
← Didn't Know|Knew It →