Answer key for a quadratic equations worksheet focusing on completing the square method.

To solve these quadratic equations by completing the square, we follow a standard process. Since the leading coefficient (the number in front of the squared term) is not always 1, the first step is usually to divide the entire equation by that number to make it easier to work with.

Here is the step-by-step verification for each problem on the worksheet:

1. $2n^2 + 7n + 3 = 0$
* Divide by 2: $n^2 + \frac{7}{2}n + \frac{3}{2} = 0$
* Move constant to right: $n^2 + \frac{7}{2}n = -\frac{3}{2}$
* Complete the square: Add $(\frac{7}{4})^2 = \frac{49}{16}$ to both sides.
* $n^2 + \frac{7}{2}n + \frac{49}{16} = -\frac{24}{16} + \frac{49}{16}$
* $(n + \frac{7}{4})^2 = \frac{25}{16}$
* Square root both sides: $n + \frac{7}{4} = \pm \frac{5}{4}$
* Solve for n:
* $n = -\frac{7}{4} + \frac{5}{4} = -\frac{2}{4} = -\frac{1}{2}$
* $n = -\frac{7}{4} - \frac{5}{4} = -\frac{12}{4} = -3$
* Answer: $n = \{-\frac{1}{2}, -3\}$

2. $2h^2 + 7h + 6 = 0$
* Divide by 2: $h^2 + \frac{7}{2}h + 3 = 0$
* Move constant: $h^2 + \frac{7}{2}h = -3$
* Complete square: Add $\frac{49}{16}$.
* $(h + \frac{7}{4})^2 = -\frac{48}{16} + \frac{49}{16} = \frac{1}{16}$
* Square root: $h + \frac{7}{4} = \pm \frac{1}{4}$
* Solve for h:
* $h = -\frac{7}{4} + \frac{1}{4} = -\frac{6}{4} = -\frac{3}{2}$
* $h = -\frac{7}{4} - \frac{1}{4} = -\frac{8}{4} = -2$
* Answer: $h = \{-\frac{3}{2}, -2\}$

3. $3k^2 + 10k + 3 = 0$
* Divide by 3: $k^2 + \frac{10}{3}k + 1 = 0$
* Move constant: $k^2 + \frac{10}{3}k = -1$
* Complete square: Add $(\frac{5}{3})^2 = \frac{25}{9}$.
* $(k + \frac{5}{3})^2 = -\frac{9}{9} + \frac{25}{9} = \frac{16}{9}$
* Square root: $k + \frac{5}{3} = \pm \frac{4}{3}$
* Solve for k:
* $k = -\frac{5}{3} + \frac{4}{3} = -\frac{1}{3}$
* $k = -\frac{5}{3} - \frac{4}{3} = -\frac{9}{3} = -3$
* Answer: $k = \{-\frac{1}{3}, -3\}$

4. $2x^2 - 21x + 27 = 0$
* Divide by 2: $x^2 - \frac{21}{2}x + \frac{27}{2} = 0$
* Move constant: $x^2 - \frac{21}{2}x = -\frac{27}{2}$
* Complete square: Add $(-\frac{21}{4})^2 = \frac{441}{16}$.
* $(x - \frac{21}{4})^2 = -\frac{216}{16} + \frac{441}{16} = \frac{225}{16}$
* Square root: $x - \frac{21}{4} = \pm \frac{15}{4}$
* Solve for x:
* $x = \frac{21}{4} + \frac{15}{4} = \frac{36}{4} = 9$
* $x = \frac{21}{4} - \frac{15}{4} = \frac{6}{4} = \frac{3}{2}$
* Answer: $x = \{9, \frac{3}{2}\}$

5. $2b^2 + 5b + 2 = 0$
* Divide by 2: $b^2 + \frac{5}{2}b + 1 = 0$
* Move constant: $b^2 + \frac{5}{2}b = -1$
* Complete square: Add $(\frac{5}{4})^2 = \frac{25}{16}$.
* $(b + \frac{5}{4})^2 = -\frac{16}{16} + \frac{25}{16} = \frac{9}{16}$
* Square root: $b + \frac{5}{4} = \pm \frac{3}{4}$
* Solve for b:
* $b = -\frac{5}{4} + \frac{3}{4} = -\frac{2}{4} = -\frac{1}{2}$
* $b = -\frac{5}{4} - \frac{3}{4} = -\frac{8}{4} = -2$
* Answer: $b = \{-\frac{1}{2}, -2\}$

6. $3c^2 - 10c + 3 = 0$
* Divide by 3: $c^2 - \frac{10}{3}c + 1 = 0$
* Move constant: $c^2 - \frac{10}{3}c = -1$
* Complete square: Add $(-\frac{5}{3})^2 = \frac{25}{9}$.
* $(c - \frac{5}{3})^2 = -\frac{9}{9} + \frac{25}{9} = \frac{16}{9}$
* Square root: $c - \frac{5}{3} = \pm \frac{4}{3}$
* Solve for c:
* $c = \frac{5}{3} + \frac{4}{3} = \frac{9}{3} = 3$
* $c = \frac{5}{3} - \frac{4}{3} = \frac{1}{3}$
* Answer: $c = \{3, \frac{1}{3}\}$

7. $2a^2 + 13a + 21 = 0$
* Divide by 2: $a^2 + \frac{13}{2}a + \frac{21}{2} = 0$
* Move constant: $a^2 + \frac{13}{2}a = -\frac{21}{2}$
* Complete square: Add $(\frac{13}{4})^2 = \frac{169}{16}$.
* $(a + \frac{13}{4})^2 = -\frac{168}{16} + \frac{169}{16} = \frac{1}{16}$
* Square root: $a + \frac{13}{4} = \pm \frac{1}{4}$
* Solve for a:
* $a = -\frac{13}{4} + \frac{1}{4} = -\frac{12}{4} = -3$
* $a = -\frac{13}{4} - \frac{1}{4} = -\frac{14}{4} = -\frac{7}{2}$
* Answer: $a = \{-3, -\frac{7}{2}\}$

8. $2z^2 - 3z - 35 = 0$
* Divide by 2: $z^2 - \frac{3}{2}z - \frac{35}{2} = 0$
* Move constant: $z^2 - \frac{3}{2}z = \frac{35}{2}$
* Complete square: Add $(-\frac{3}{4})^2 = \frac{9}{16}$.
* $(z - \frac{3}{4})^2 = \frac{280}{16} + \frac{9}{16} = \frac{289}{16}$
* Square root: $z - \frac{3}{4} = \pm \frac{17}{4}$
* Solve for z:
* $z = \frac{3}{4} + \frac{17}{4} = \frac{20}{4} = 5$
* $z = \frac{3}{4} - \frac{17}{4} = -\frac{14}{4} = -\frac{7}{2}$
* Answer: $z = \{5, -\frac{7}{2}\}$

9. $3r^2 + 7r + 4 = 0$
* Divide by 3: $r^2 + \frac{7}{3}r + \frac{4}{3} = 0$
* Move constant: $r^2 + \frac{7}{3}r = -\frac{4}{3}$
* Complete square: Add $(\frac{7}{6})^2 = \frac{49}{36}$.
* $(r + \frac{7}{6})^2 = -\frac{48}{36} + \frac{49}{36} = \frac{1}{36}$
* Square root: $r + \frac{7}{6} = \pm \frac{1}{6}$
* Solve for r:
* $r = -\frac{7}{6} + \frac{1}{6} = -\frac{6}{6} = -1$
* $r = -\frac{7}{6} - \frac{1}{6} = -\frac{8}{6} = -\frac{4}{3}$
* Answer: $r = \{-1, -\frac{4}{3}\}$

10. $3y^2 - 8y + 4 = 0$
* Divide by 3: $y^2 - \frac{8}{3}y + \frac{4}{3} = 0$
* Move constant: $y^2 - \frac{8}{3}y = -\frac{4}{3}$
* Complete square: Add $(-\frac{4}{3})^2 = \frac{16}{9}$.
* $(y - \frac{4}{3})^2 = -\frac{12}{9} + \frac{16}{9} = \frac{4}{9}$
* Square root: $y - \frac{4}{3} = \pm \frac{2}{3}$
* Solve for y:
* $y = \frac{4}{3} + \frac{2}{3} = \frac{6}{3} = 2$
* $y = \frac{4}{3} - \frac{2}{3} = \frac{2}{3}$
* Answer: $y = \{2, \frac{2}{3}\}$

Final Answer:
n = {-½, -3}
h = {-³/₂, -2}
k = {-⅓, -3}
x = {9, ³/₂}
b = {-½, -2}
c = {3, ⅓}
a = {-3, -⁷/₂}
z = {5, -⁷/₂}
r = {-1, -⁴/₃}
y = {2, ²/₃}