Finite Math Examples

$\frac{- 1}{{(x + 3)}^{2}} = - \frac{1}{x^{2} + 6 x + 9}$

Step 1

Move the negative in front of the fraction.

$- \frac{1}{{(x + 3)}^{2}} = - \frac{1}{x^{2} + 6 x + 9}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$- \frac{1}{{(x + 3)}^{2}} = - \frac{1}{x^{2} + 6 x + 3^{2}}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 2.3

Rewrite the polynomial.

$- \frac{1}{{(x + 3)}^{2}} = - \frac{1}{x^{2} + 2 \cdot x \cdot 3 + 3^{2}}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$- \frac{1}{{(x + 3)}^{2}} = - \frac{1}{{(x + 3)}^{2}}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 49.5, - 49.2, - 49, - 48.9, - 48.5, - 48.4, - 48, - 47.9, - 47.7, - 47.6, - 47.5, - 47.3, - 47.1, - 47, - 46.3\overline{9}, - 46.0\overline{9}, - 45.8\overline{9}, - 45.7\overline{9}, - 45.5\overline{9}, - 45.4\overline{9}, - 45.0\overline{9}, - 45, - 43.4\overline{9}, - 43.3\overline{9}, - 43.2\overline{9}, - 43.1\overline{9}, - 42.8\overline{9}, - 42.7\overline{9}, - 41.6\overline{9}, - 41.5\overline{9}, - 41.4\overline{9}, - 41.2\overline{9}, - 41, - 40.6\overline{9}, - 40.5\overline{9}, - 40.4\overline{9}, - 40.3\overline{9}, - 40.2\overline{9}, - 40, - 39.5\overline{9}, - 39.4\overline{9}, - 39.3\overline{9}, - 39.2\overline{9}, - 38.7\overline{9}, - 38.5\overline{9}, - 38.3\overline{9}, - 38.2\overline{9}, - 38.1\overline{9}, - 38.0\overline{9}, - 38, - 37.8\overline{9}, - 37.7\overline{9}, - 37.3\overline{9}, - 37.2\overline{9}, - 37.0\overline{9}, - 37, - 36.7\overline{9}, - 36.6\overline{9}, - 36.3\overline{9}, - 36.2\overline{9}, - 35.8\overline{9}, - 35.7\overline{9}, - 35.4\overline{9}, - 35.3\overline{9}, - 34.3\overline{9}, - 34.2\overline{9}, - 34.0\overline{9}, - 34, - 33.2\overline{9}, - 33.1\overline{9}, - 33.0\overline{9}, - 32.8\overline{9}, - 32.5\overline{9}, - 32.4\overline{9}, - 32.2\overline{9}, - 32.0\overline{9}, - 32, - 31.7\overline{9}, - 31.5\overline{9}, - 31.3\overline{9}, - 31.2\overline{9}, - 31.1\overline{9}, - 31.0\overline{9}, - 31, - 30.8\overline{9}, - 30.7\overline{9}, - 30.6\overline{9}, - 30.5\overline{9}, - 29.5\overline{9}, - 29.1\overline{9}, - 28.6\overline{9}, - 28.5\overline{9}, - 28.4\overline{9}, - 28.3\overline{9}, - 28.2\overline{9}, - 28.1\overline{9}, - 28.0\overline{9}, - 28, - 27.8\overline{9}, - 27.6\overline{9}, - 27.1\overline{9}, - 27.0\overline{9}, - 26.8\overline{9}, - 26.6\overline{9}, - 26.5\overline{9}, - 26.4\overline{9}, - 26.3\overline{9}, - 26.2\overline{9}, - 26.1\overline{9}, - 25.8\overline{9}, - 25.7\overline{9}, - 25.5\overline{9}, - 25.4\overline{9}, - 25.3\overline{9}, - 25.2\overline{9}, - 25, - 24.8\overline{9}, - 24.7\overline{9}, - 24.6\overline{9}, - 24.3\overline{9}, - 24.2\overline{9}, - 23.8\overline{9}, - 23.7\overline{9}, - 23.6\overline{9}, - 23.5\overline{9}, - 23.1\overline{9}, - 23.0\overline{9}, - 23, - 22.8\overline{9}, - 22.7\overline{9}, - 22.6\overline{9}, - 22.1\overline{9}, - 22.0\overline{9}, - 22, - 21.4\overline{9}, - 21.3\overline{9}, - 21.2\overline{9}, - 21.1\overline{9}, - 21, - 20.8\overline{9}, - 20.7\overline{9}, - 20.6\overline{9}, - 20.5\overline{9}, - 20.2\overline{9}, - 20.1\overline{9}, - 20.0\overline{9}, - 20, - 19.8\overline{9}, - 19.7\overline{9}, - 19.5\overline{9}, - 19.4\overline{9}, - 19.3\overline{9}, - 19.2\overline{9}, - 18.7\overline{9}, - 18.6\overline{9}, - 18.4\overline{9}, - 18.3\overline{9}, - 18.1\overline{9}, - 18, - 17.8\overline{9}, - 17.5\overline{9}, - \frac{35}{2}, - 17.2\overline{9}, - 17, - 16.5\overline{9}, - \frac{33}{2}, - 16.0\overline{9}, - 16, - 15.5\overline{9}, - \frac{31}{2}, - 15, - 14.7\overline{9}, - 14.6\overline{9}, - 14.5\overline{9}, - \frac{29}{2}, - 14.2\overline{9}, - 13.8\overline{9}, - 13.7\overline{9}, - 13.6\overline{9}, - 13.5\overline{9}, - \frac{27}{2}, - 13.3\overline{9}, - 13.2\overline{9}, - 13.0\overline{9}, - 13, - 12.8\overline{9}, - 12.6\overline{9}, - 12.5\overline{9}, - \frac{62}{5}, - 12.2\overline{9}, - 11.6\overline{9}, - \frac{58}{5}, - \frac{23}{2}, - \frac{57}{5}, - 11.0\overline{9}, - 11, - \frac{54}{5}, - 10.6\overline{9}, - \frac{21}{2}, - 10.2\overline{9}, - \frac{51}{5}, - 10.0\overline{9}, - 9.9, - \frac{49}{5}, - \frac{48}{5}, - \frac{47}{5}, - 9.3, - 9.1, - 8.9, - \frac{44}{5}, - \frac{43}{5}, - \frac{17}{2}, - 8.3, - 8.1, - 7.9, - \frac{39}{5}, - \frac{38}{5}, - \frac{37}{5}, - \frac{36}{5}, - 7.1, - 6.9, - \frac{34}{5}, - \frac{33}{5}, - \frac{32}{5}, - 6.3, - 6, - 5.9, - 5.7, - \frac{11}{2}, - \frac{27}{5}, - 5.3, - \frac{26}{5}, - 5.1, - 5, - \frac{24}{5}, - 4.7, - \frac{23}{5}, - \frac{22}{5}, - \frac{19}{5}, - \frac{18}{5}, - \frac{7}{2}, - 3.3, - 3.1, - 2.9, - \frac{14}{5}, - 2.7, - \frac{13}{5}, - \frac{5}{2}, - \frac{12}{5}, - 2.3, - \frac{11}{5}, - 2.1, - \frac{9}{5}, - \frac{8}{5}, - \frac{3}{2}, - 1.3, - 1.1, - 1, - \frac{4}{5}, - 0.7, - \frac{3}{5}, - \frac{2}{5}, - 0.3, 0, 0.1, \frac{1}{5}, \frac{1}{2}, \frac{3}{5}, 0.7, 0.9, 1, 1.1, \frac{6}{5}, 1.3, \frac{7}{5}, \frac{3}{2}, \frac{8}{5}, 1.7, \frac{9}{5}, 1.9, 2.1, \frac{11}{5}, 2.7, \frac{14}{5}, 2.9, 3, 3.1, \frac{16}{5}, 3.3, \frac{17}{5}, \frac{7}{2}, 3.7, \frac{19}{5}, 3.9, 4, 4.1, \frac{21}{5}, 4.3, \frac{9}{2}, \frac{23}{5}, 4.7, \frac{11}{2}, \frac{28}{5}, 5.9, 6, 6.1, \frac{31}{5}, 6.3, \frac{32}{5}, \frac{13}{2}, 6.7, \frac{34}{5}, 7, \frac{15}{2}, \frac{38}{5}, \frac{39}{5}, 7.9, 8.1, \frac{41}{5}, 8.7, 8.9, 9.1, \frac{46}{5}, \frac{19}{2}, \frac{48}{5}, \frac{52}{5}, \frac{21}{2}, \frac{53}{5}, 10.7, 11.1, \frac{56}{5}, \frac{59}{5}, 12, 12.2, 12.3, \frac{62}{5}, \frac{25}{2}, 12.6, 12.8, 12.9, 13.1, 13.3, \frac{27}{2}, 13.6, 13.7, 13.8, 13.9, 14, 14.1, 14.2, 14.3, 14.4, \frac{29}{2}, 14.7, 14.8, 14.9, 15.2, 15.3, 15.4, \frac{31}{2}, 15.7, 15.9, 16, 16.2, 16.3, 16.4, 16.7, 16.8, 17, 17.2, 17.3, 17.4, \frac{35}{2}, 17.6, 17.7, 18.3, 18.4, 18.5, 19.2, 19.3, 19.8, 19.9, 20, 20.1, 20.4, 20.5, 20.6, 20.7, 21, 21.1, 21.2, 21.3, 21.6, 21.7, 21.8, 21.9, 22.2, 22.3, 22.7, 22.8, 23, 23.1, 23.7, 23.9, 24, 24.1, 24.2, 25.8, 25.9, 26, 26.1, 26.4, 26.5, 26.9, 27, 27.3, 27.4, 27.9, 28.1, 28.7, 28.8, 29.1, 29.2, 29.3, 29.4, 29.5, 29.6, 29.7, 30, 30.1, 30.2, 30.3, 30.6, 30.7, 30.8, 30.9, 31, 31.1, 31.2, 31.3, 31.4, 31.5, 31.6, 31.7, 31.8, 31.9, 32.1, 32.2, 32.3, 32.5, 32.6, 33, 33.1, 33.2, 33.3, 33.4, 33.5, 33.6, 34.1, 34.2, 34.6, 34.7, 34.8, 34.9, 35, 35.1, 35.3, 35.5, 35.6, 36, 36.1, 36.2, 36.3, 36.4, 36.5, 36.6, 36.7, 38, 38.1, 38.4, 38.5, 38.9, 39, 39.3, 39.4, 39.5, 39.6, 39.8, 39.9, 40, 40.2, 40.5, 40.6, 41.2, 41.3, 41.7, 41.8, 42, 42.1, 42.2, 42.5, 43.1, 43.2, 44.2, 44.3, 44.4, 44.5, 44.9, 45, 45.4, 45.5, 45.6, 45.7, 45.8, 45.9, 46.2, 46.3, 47.1, 47.2, 47.6, 47.7, 48.4, 48.7, 48.8, 48.9, 49, 49.1, 49.2, 49.4, 49.5, 49.8, 49.9$

Step 4