Solve Quadratic Equations with Complex Solutions 2

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Question 1 of 4

Solve

$2 x^{2} + 3 x + 9 = 0$ $2x^2+3x+9=0$

1. $- \frac{3}{4} + \frac{3 i \sqrt{7}}{4}$ $-3/4 + (3isqrt(7))/4$ and $- \frac{3}{4} - \frac{3 i \sqrt{7}}{4}$ $-3/4 - (3isqrt(7))/4$
2. $- \frac{3}{2} + \frac{3 i \sqrt{7}}{2}$ $-3/2 + (3isqrt(7))/2$ and $- \frac{3}{2} - \frac{3 i \sqrt{7}}{2}$ $-3/2 - (3isqrt(7))/2$
3. $- \frac{3}{4} + 3 i \sqrt{7}$ $-3/4 + 3isqrt(7)$ and $- \frac{3}{4} - 3 i \sqrt{7}$ $-3/4 - 3isqrt(7)$
4. $- \frac{3}{2} + \frac{3 i \sqrt{7}}{4}$ $-3/2 + (3isqrt(7))/4$ and $- \frac{3}{2} - \frac{3 i \sqrt{7}}{4}$ $-3/2 - (3isqrt(7))/4$

Question 2 of 4

Solve

$2 x^{2} + 10 x + 17 = 0$ $2x^2+10x+17=0$

1. $- \frac{5}{2} + \frac{3 i}{2}$ $-5/2 +(3i)/2$ and $- \frac{5}{2} - \frac{3 i}{2}$ $-5/2-(3i)/2$
2. $- \frac{5}{4} + \frac{3 i}{2}$ $-5/4 +(3i)/2$ and $- \frac{5}{4} - \frac{3 i}{2}$ $-5/4-(3i)/2$
3. $- \frac{5}{2} + 3 i$ $-5/2 +3i$ and $- \frac{5}{2} - 3 i$ $-5/2-3i$
4. $- 5 + \frac{3 i}{2}$ $-5 +(3i)/2$ and $- 5 - \frac{3 i}{2}$ $-5-(3i)/2$

Question 3 of 4

Solve

$3 x^{2} + 2 x + 11 = 0$ $3x^2+2x+11=0$

1. $- 2 + 8 i \sqrt{2}$ $-2+8isqrt(2)$ and $- 2 - 8 i \sqrt{2}$ $-2-8isqrt(2)$
2. $- \frac{1}{6} + \frac{4 i \sqrt{2}}{6}$ $-1/6+(4isqrt(2))/6$ and $- \frac{1}{6} - \frac{4 i \sqrt{2}}{6}$ $-1/6-(4isqrt(2))/6$
3. $2 + 8 i \sqrt{2}$ $2+8isqrt(2)$ and $2 - 8 i \sqrt{2}$ $2-8isqrt(2)$
4. $- \frac{1}{3} + \frac{4 i \sqrt{2}}{3}$ $-1/3+(4isqrt(2))/3$ and $- \frac{1}{3} - \frac{4 i \sqrt{2}}{3}$ $-1/3-(4isqrt(2))/3$

Question 4 of 4

Solve

$x^2-2sqrt2x+6=0$

1. $sqrt2+2i$ and $sqrt2-2i$
2. $sqrt2+i$ and $sqrt2-i$
3. $2sqrt2+2i$ and $2sqrt2-2i$
4. $2isqrt2$ and $-2isqrt2$

$x =$ $x=$	$\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $(color(red)(-b)\+-sqrt(color(red)(b)^2-4color(blue)(a)color(green)(c)))/(2color(blue)(a))$	Substitute in $a = 2$ $a=2$ , $b = 3$ $b=3$ , and $c = 9$ $c=9$ .
$x =$ $x=$	$\frac{- 3 \pm \sqrt{3^{2} - 4 (2) (9)}}{2 (2)}$ $(-color(red)(3)\+-sqrt(color(red)(3)^2-4color(blue)((2))color(green)((9))))/(2color(blue)((2)))$	Simplify
$x =$ $x=$	$\frac{- 3 \pm \sqrt{3^{2} - 4 (2) (9)}}{4}$ $(-3\+-sqrt(3^2-4(2)(9)))/(4)$	Simplify $3^{2}$ $3^2$
$x =$ $x=$	$\frac{- 3 \pm \sqrt{9 - 4 (2) (9)}}{4}$ $(-3\+-sqrt(9-4(2)(9)))/(4)$	Multiply $- 4 (2) (9)$ $-4(2)(9)$ .
$x =$ $x=$	$\frac{- 3 \pm \sqrt{9 - 72}}{4}$ $(-3\+-sqrt(9-72))/(4)$	Subtract under the root.
$x =$ $x=$	$\frac{- 3 \pm \sqrt{- 63}}{4}$ $(-3\+-sqrt(-63))/(4)$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$x =$ $x=$	$\frac{- 3 \pm \sqrt{- 1} \times \sqrt{63}}{4}$ $(-3\+-sqrt(-1)timessqrt(63))/(4)$	Replace the $\sqrt{- 1}$ $sqrt(-1)$ remember $\sqrt{- 1} = i$ $sqrt(-1)=i$
$x =$ $x=$	$\frac{- 3 \pm i \times \sqrt{9} \times \sqrt{7}}{4}$ $(-3\+-itimessqrt(9)timessqrt(7))/(4)$	Simplify the root using $\sqrt{63} = \sqrt{9} \times \sqrt{7}$ $sqrt(63) = sqrt(9)timessqrt(7)$ .
$x =$ $x=$	$\frac{- 3 \pm i \times 3 \times \sqrt{7}}{4}$ $(-3\+-itimes3timessqrt(7))/(4)$	Simplify the root.
$x =$ $x=$	$\frac{- 3 \pm 3 i \sqrt{7}}{4}$ $(-3\+-3isqrt(7))/(4)$	Rearrange $3 \times i \times \sqrt{7}$ $3timesitimessqrt(7)$ so that the $i$ $i$ is in between the constant and the root.
$x =$ $x=$	$- \frac{3}{4} \pm 3 i \frac{\sqrt{7}}{4}$ $-3/4 \+-3isqrt(7)/4$	Divide both terms in the numerator by $4$ $4$ .
	$- \frac{3}{4} + 3 i \frac{\sqrt{7}}{4}$ $-3/4 + 3isqrt(7)/4$ and $- \frac{3}{4} - 3 i \frac{\sqrt{7}}{4}$ $-3/4 – 3isqrt(7)/4$	Break into the two solutions.

$x =$ $x=$	$\frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $(color(red)(-b)\+-sqrt(color(red)(b)^2-4color(blue)(a)color(green)(c)))/(2color(blue)(a))$	Substitute in $a = 2$ $a=2$ , $b = 10$ $b=10$ , and $c = 17$ $c=17$ .
$x =$ $x=$	$\frac{- 10 \pm \sqrt{10^{2} - 4 (2) (17)}}{2 (2)}$ $(-color(red)(10)\+-sqrt(color(red)(10)^2-4color(blue)((2))color(green)((17))))/(2color(blue)((2)))$	Simplify
$x =$ $x=$	$\frac{- 10 \pm \sqrt{10^{2} - 4 (2) (17)}}{4}$ $(-10\+-sqrt(10^2-4(2)(17)))/(4)$	Simplify $10^{2}$ $10^2$
$x =$ $x=$	$\frac{- 10 \pm \sqrt{100 - 4 (2) (17)}}{4}$ $(-10\+-sqrt(100-4(2)(17)))/(4)$	Multiply $- 4 (2) (17)$ $-4(2)(17)$ .
$x =$ $x=$	$\frac{- 10 \pm \sqrt{100 - 136}}{4}$ $(-10\+-sqrt(100-136))/(4)$	Subtract under the root.
$x =$ $x=$	$\frac{- 10 \pm \sqrt{- 36}}{4}$ $(-10\+-sqrt(-36))/(4)$	Separate out the $\sqrt{- 1}$ $sqrt(-1)$ from the root.
$x =$ $x=$	$\frac{- 10 \pm \sqrt{- 1} \times \sqrt{36}}{4}$ $(-10\+-sqrt(-1)timessqrt(36))/(4)$	Replace the $\sqrt{- 1}$ $sqrt(-1)$ remember $\sqrt{- 1} = i$ $sqrt(-1)=i$
$x =$ $x=$	$\frac{- 10 \pm i \times 6}{4}$ $(-10\+-itimes6)/(4)$	Simplify the root.
$x =$ $x=$	$\frac{- 10 \pm 6 i}{4}$ $(-10\+-6i)/(4)$	Rearrange $i \times 6$ $itimes6$ so that the $i$ $i$ is in between the constant and the root.
$x =$ $x=$	$- \frac{10}{4} \pm \frac{6 i}{4}$ $-10/4 \+-(6i)/4$	Divide both terms in the numerator by $4$ $4$ .
$x =$ $x=$	$- \frac{5}{2} \pm \frac{3 i}{2}$ $-5/2 \+-(3i)/2$	Simplify.
	$- \frac{5}{2} + \frac{3 i}{2}$ $-5/2 +(3i)/2$ and $- \frac{5}{2} - \frac{3 i}{2}$ $-5/2-(3i)/2$	Break into the two solutions.

$x=$	$(color(red)(-b)\+-sqrt(color(red)(b)^2-4color(blue)(a)color(green)(c)))/(2color(blue)(a))$	Substitute in $a=3$ , $b=2$ , and $c=11$ .
$x=$	$(-color(red)(2)\+-sqrt(color(red)(2)^2-4color(blue)((3))color(green)((11))))/(2color(blue)((3)))$	Simplify
$x=$	$(-2\+-sqrt(2^2-4(3)(11)))/(6)$	Simplify $2^2$
$x=$	$(-2\+-sqrt(4-4(3)(11)))/(6)$	Multiply $-4(3)(11)$ .
$x=$	$(-2\+-sqrt(4-132))/(6)$	Subtract under the root.
$x=$	$(-2\+-sqrt(-128))/(6)$	Separate out the $sqrt(-1)$ from the root.
$x=$	$(-2\+-sqrt(-1)timessqrt(128))/(6)$	Replace the $sqrt(-1)$ remember $sqrt(-1)=i$
$x=$	$(-2\+-itimessqrt(64)timessqrt(2))/(6)$	Simplify the root using $sqrt(128) = sqrt(64)timessqrt(2)$ .
$x=$	$(-2\+-itimes8timessqrt(2))/(6)$	Simplify the root.
$x=$	$(-2\+-8isqrt(2))/(6)$	Rearrange $itimes8timessqrt(2)$ so that the $i$ is in between the constant and the root.
$x=$	$-2/6 \+-(8isqrt(2))/6$	Divide both terms in the numerator by $6$ .
	$-1/3 \+-(4isqrt(2))/3$	Simplify.
	$-1/3+(4isqrt(2))/3$ and $-1/3-(4isqrt(2))/3$	Break into the two solutions.

$x=$	$(color(red)(-b)\+-sqrt(color(red)(b)^2-4color(blue)(a)color(green)(c)))/(2color(blue)(a))$	Substitute in $a=1$ , $b=-2sqrt2$ , and $c=6$ .
$x=$	$(-color(red)(-2sqrt2)\+-sqrt(color(red)(-2sqrt2)^2-4color(blue)((1))color(green)((6))))/(2color(blue)((1)))$	Simplify
$x=$	$(-(-2sqrt2)\+-sqrt((-2sqrt2)^2-4(1)(6)))/(2)$	Simplify $(-2sqrt2)^2$ .
$x=$	$(2sqrt2\+-sqrt(8-4(1)(6)))/(2)$	Multiply $-4(1)(6)$ .
$x=$	$(2sqrt2\+-sqrt(8-24))/(2)$	Subtract under the root.
$x=$	$(2sqrt2\+-sqrt(-16))/(2)$	Separate out the $sqrt(-1)$ from the root.
$x=$	$(2sqrt2\+-sqrt(-1)timessqrt(16))/(2)$	Replace the $sqrt(-1)$ remember $sqrt(-1)=i$
$x=$	$(2sqrt2\+-itimessqrt(16))/(2)$	Simplify the root.
$x=$	$(2sqrt2\+-4i)/(2)$	Rearrange $itimessqrt(16)$ so that the $i$ is on the right-hand side in this term.
$x=$	$(2sqrt2)/2\+-(4i)/2$	Divide both terms in the numerator by $2$ .
$x=$	$sqrt2\+-2i$
	$sqrt2+2i$ and $sqrt2-2i$	Break into the two solutions.

Solve Quadratic Equations with Complex Solutions 2

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