Complex Numbers and Quadratic Equations Exercise 5.3

Solve each of the following equations.
1. $x^{2}+3=0$
$a=1$, $b=0$, $c=3$
$x^{2}=-3$
$x=\pm \sqrt{-3}$
$x=\pm i\sqrt{3}$

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2. $2x^{2}+x+1=0$
$a=2$, $b=1$, $c=1$
$b^{2}-4ac$ $=1-8=-7$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{-1\pm i\sqrt{7}}{4}$

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3. $x^{2}+3x+9=0$
$a=1$, $b=3$, $c=9$
$b^{2}-4ac$ $=9-36=-27$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{-3\pm i\sqrt{27}}{2}$
= $\large\frac{-3\pm i3\sqrt{3}}{2}$

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4.  $-x^{2}+x-2=0$
$a=-1$, $b=1$, $c=-2$
$b^{2}-4ac$ $=1-8=-7$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{-1\pm i\sqrt{7}}{-2}$

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5. $x^{2}+3x+5=0$
$a=1$, $b=3$, $c=5$
$b^{2}-4ac$ $=9-20=-11$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{-3\pm i\sqrt{11}}{2}$

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6. $x^{2}-x+2=0$
$a=1$, $b=-1$, $c=2$
$b^{2}-4ac$ $=1-8=-7$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{1\pm i\sqrt{7}}{2}$

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7. $\sqrt{2}x^{2}+x+\sqrt{2}=0$
$a=\sqrt{2}$, $b=1$, $c=\sqrt{2}$
$b^{2}-4ac$ $=1-8=-7$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{-1\pm i\sqrt{7}}{2\sqrt{2}}$

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8.  $\sqrt{3}x^{2}-\sqrt{2}x+3\sqrt{3}=0$
$a=\sqrt{3}$, $b=-\sqrt{2}$, $c=3\sqrt{3}$
$b^{2}-4ac$ $=2-36=-34$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{\sqrt{2}\pm i\sqrt{34}}{2\sqrt{3}}$

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9. $x^{2}+x+\frac{1}{\sqrt{2}}=0$
$a=1$, $b=1$, $c=\frac{1}{\sqrt{2}}$
$b^{2}-4ac$ $=1-\frac{4}{\sqrt{2}}$ $=1-2\sqrt{2}$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{-1\pm i\sqrt{2\sqrt{2}-1}}{2}$

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10. $x^{2}+\frac{x}{\sqrt{2}}+1=0$
$a=1$, $b=\frac{1}{\sqrt{2}}$, $c=1$
$b^{2}-4ac$ $=\frac{1}{2}-4$ $=-\frac{7}{2}$
$x=\frac{-b\pm i\sqrt{4ac-b^{2}}}{2a}$
= $\large\frac{-\frac{1}{\sqrt{2}}\pm i\sqrt{\frac{7}{2}}}{2}$
= $\large\frac{-1\pm i\sqrt{7}}{2\sqrt{2}}$

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