Question

1. $6x^2 - 34 = -5x$
2. $10n^2 + 7 = -3n$
3. $5n^2 + 10n = -10$
4. $5a^2 = 5a + 14$
5. $4p^2 - 36 = 0$
6. $5r^2 + 4r = 57$
7. $10p^2 = 6p + 11$
8. $9x^2 = -2x + 20$

Explanation:

Problem 7: $6x^2 - 34 = -5x$

Step1: Rearrange to standard form

$6x^2 +5x -34 = 0$

Step2: Apply quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=6$, $b=5$, $c=-34$
$$x=\frac{-5\pm\sqrt{5^2-4(6)(-34)}}{2(6)}=\frac{-5\pm\sqrt{25+816}}{12}=\frac{-5\pm\sqrt{841}}{12}=\frac{-5\pm29}{12}$$

Step3: Calculate two solutions

$x_1=\frac{-5+29}{12}=\frac{24}{12}=2$, $x_2=\frac{-5-29}{12}=\frac{-34}{12}=-\frac{17}{6}$

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Problem 8: $10n^2 +7 = -3n$

Step1: Rearrange to standard form

$10n^2 +3n +7 = 0$

Step2: Apply quadratic formula $n=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=10$, $b=3$, $c=7$
$$n=\frac{-3\pm\sqrt{3^2-4(10)(7)}}{2(10)}=\frac{-3\pm\sqrt{9-280}}{20}=\frac{-3\pm\sqrt{-271}}{20}$$

Step3: Simplify complex solutions

$n_1=\frac{-3+i\sqrt{271}}{20}$, $n_2=\frac{-3-i\sqrt{271}}{20}$

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Problem 9: $5n^2 +10n = -10$

Step1: Rearrange to standard form

$5n^2 +10n +10 = 0$

Step2: Simplify equation

Divide by 5: $n^2 +2n +2 = 0$

Step3: Apply quadratic formula $n=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=1$, $b=2$, $c=2$
$$n=\frac{-2\pm\sqrt{2^2-4(1)(2)}}{2(1)}=\frac{-2\pm\sqrt{4-8}}{2}=\frac{-2\pm\sqrt{-4}}{2}=\frac{-2\pm2i}{2}$$

Step4: Simplify solutions

$n_1=-1+i$, $n_2=-1-i$

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Problem 10: $5a^2 = 5a +14$

Step1: Rearrange to standard form

$5a^2 -5a -14 = 0$

Step2: Apply quadratic formula $a=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=5$, $b=-5$, $c=-14$
$$a=\frac{5\pm\sqrt{(-5)^2-4(5)(-14)}}{2(5)}=\frac{5\pm\sqrt{25+280}}{10}=\frac{5\pm\sqrt{305}}{10}$$

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Problem 11: $4p^2 -36 = 0$

Step1: Isolate squared term

$4p^2=36$

Step2: Solve for $p^2$

$p^2=\frac{36}{4}=9$

Step3: Take square root

$p=\pm3$

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Problem 12: $5r^2 +4r = 57$

Step1: Rearrange to standard form

$5r^2 +4r -57 = 0$

Step2: Apply quadratic formula $r=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=5$, $b=4$, $c=-57$
$$r=\frac{-4\pm\sqrt{4^2-4(5)(-57)}}{2(5)}=\frac{-4\pm\sqrt{16+1140}}{10}=\frac{-4\pm\sqrt{1156}}{10}=\frac{-4\pm34}{10}$$

Step3: Calculate two solutions

$r_1=\frac{-4+34}{10}=\frac{30}{10}=3$, $r_2=\frac{-4-34}{10}=\frac{-38}{10}=-\frac{19}{5}$

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Problem 13: $10p^2 = 6p +11$

Step1: Rearrange to standard form

$10p^2 -6p -11 = 0$

Step2: Apply quadratic formula $p=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=10$, $b=-6$, $c=-11$
$$p=\frac{6\pm\sqrt{(-6)^2-4(10)(-11)}}{2(10)}=\frac{6\pm\sqrt{36+440}}{20}=\frac{6\pm\sqrt{476}}{20}=\frac{6\pm2\sqrt{119}}{20}=\frac{3\pm\sqrt{119}}{10}$$

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Problem 14: $9x^2 = -2x +20$

Step1: Rearrange to standard form

$9x^2 +2x -20 = 0$

Step2: Apply quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=9$, $b=2$, $c=-20$
$$x=\frac{-2\pm\sqrt{2^2-4(9)(-20)}}{2(9)}=\frac{-2\pm\sqrt{4+720}}{18}=\frac{-2\pm\sqrt{724}}{18}=\frac{-2\pm2\sqrt{181}}{18}=\frac{-1\pm\sqrt{181}}{9}$$

Answer:

1. $x=2$ and $x=-\frac{17}{6}$
2. $n=\frac{-3+i\sqrt{271}}{20}$ and $n=\frac{-3-i\sqrt{271}}{20}$
3. $n=-1+i$ and $n=-1-i$
4. $a=\frac{5+\sqrt{305}}{10}$ and $a=\frac{5-\sqrt{305}}{10}$
5. $p=3$ and $p=-3$
6. $r=3$ and $r=-\frac{19}{5}$
7. $p=\frac{3+\sqrt{119}}{10}$ and $p=\frac{3-\sqrt{119}}{10}$
8. $x=\frac{-1+\sqrt{181}}{9}$ and $x=\frac{-1-\sqrt{181}}{9}$