Algebra 1 Worksheets | Quadratic Functions Worksheets - Free Printable

Here are the step-by-step solutions for each quadratic equation using the method of completing the square.

1) $z^2 - 18z + 27 = -9$
* Move the constant to the right side: $z^2 - 18z = -36$
* Take half of the middle coefficient (-18), which is -9, and square it ($(-9)^2 = 81$). Add 81 to both sides.
* $z^2 - 18z + 81 = -36 + 81$
* Factor the left side: $(z - 9)^2 = 45$
* Take the square root: $z - 9 = \pm\sqrt{45}$
* Simplify $\sqrt{45}$ to $3\sqrt{5}$.
* $z = 9 \pm 3\sqrt{5}$

2) $p^2 + 18p = -77$
* Take half of 18 (which is 9) and square it ($81$). Add 81 to both sides.
* $p^2 + 18p + 81 = -77 + 81$
* Factor the left side: $(p + 9)^2 = 4$
* Take the square root: $p + 9 = \pm 2$
* Solve for p:
* $p = -9 + 2 = -7$
* $p = -9 - 2 = -11$

3) $12s^2 - 68s = -40$
* Divide everything by 4 to simplify: $3s^2 - 17s = -10$
* Divide by 3 to make the leading coefficient 1: $s^2 - \frac{17}{3}s = -\frac{10}{3}$
* Half of $-\frac{17}{3}$ is $-\frac{17}{6}$. Square it: $\frac{289}{36}$. Add to both sides.
* $s^2 - \frac{17}{3}s + \frac{289}{36} = -\frac{10}{3} + \frac{289}{36}$
* Convert $-\frac{10}{3}$ to $-\frac{120}{36}$. Right side becomes $\frac{169}{36}$.
* $(s - \frac{17}{6})^2 = \frac{169}{36}$
* Take square root: $s - \frac{17}{6} = \pm \frac{13}{6}$
* $s = \frac{17}{6} \pm \frac{13}{6}$
* $s = \frac{30}{6} = 5$
* $s = \frac{4}{6} = \frac{2}{3}$

4) $6z^2 - 37z + 4 = -1$
* Move constant: $6z^2 - 37z = -5$
* Divide by 6: $z^2 - \frac{37}{6}z = -\frac{5}{6}$
* Half of $-\frac{37}{6}$ is $-\frac{37}{12}$. Square it: $\frac{1369}{144}$. Add to both sides.
* Right side: $-\frac{5}{6} + \frac{1369}{144} = -\frac{120}{144} + \frac{1369}{144} = \frac{1249}{144}$
* $(z - \frac{37}{12})^2 = \frac{1249}{144}$
* $z - \frac{37}{12} = \pm \frac{\sqrt{1249}}{12}$
* $z = \frac{37 \pm \sqrt{1249}}{12}$

5) $g^2 + 19g = 72$
* Half of 19 is $\frac{19}{2}$. Square it: $\frac{361}{4}$. Add to both sides.
* $g^2 + 19g + \frac{361}{4} = 72 + \frac{361}{4}$
* Right side: $\frac{288}{4} + \frac{361}{4} = \frac{649}{4}$
* $(g + \frac{19}{2})^2 = \frac{649}{4}$
* $g + \frac{19}{2} = \pm \frac{\sqrt{649}}{2}$
* $g = \frac{-19 \pm \sqrt{649}}{2}$

6) $x^2 + x - 2 = 0$
* Move constant: $x^2 + x = 2$
* Half of 1 is $\frac{1}{2}$. Square it: $\frac{1}{4}$. Add to both sides.
* $x^2 + x + \frac{1}{4} = 2 + \frac{1}{4} = \frac{9}{4}$
* $(x + \frac{1}{2})^2 = \frac{9}{4}$
* $x + \frac{1}{2} = \pm \frac{3}{2}$
* $x = \frac{3}{2} - \frac{1}{2} = 1$
* $x = -\frac{3}{2} - \frac{1}{2} = -2$

7) $6w^2 - 48w - 50 = 0$
* Move constant: $6w^2 - 48w = 50$
* Divide by 2 to simplify: $3w^2 - 24w = 25$
* Divide by 3: $w^2 - 8w = \frac{25}{3}$
* Half of -8 is -4. Square it: 16. Add to both sides.
* $w^2 - 8w + 16 = \frac{25}{3} + 16 = \frac{25}{3} + \frac{48}{3} = \frac{73}{3}$
* $(w - 4)^2 = \frac{73}{3}$
* $w - 4 = \pm \sqrt{\frac{73}{3}}$
* Rationalize denominator: $\sqrt{\frac{73}{3}} = \frac{\sqrt{219}}{3}$
* $w = 4 \pm \frac{\sqrt{219}}{3}$ or $\frac{12 \pm \sqrt{219}}{3}$

8) $20h^2 - 48h = 6$
* Divide by 2: $10h^2 - 24h = 3$
* Divide by 10: $h^2 - \frac{24}{10}h = \frac{3}{10} \rightarrow h^2 - \frac{12}{5}h = \frac{3}{10}$
* Half of $-\frac{12}{5}$ is $-\frac{6}{5}$. Square it: $\frac{36}{25}$. Add to both sides.
* Right side: $\frac{3}{10} + \frac{36}{25}$. Common denominator is 50.
* $\frac{15}{50} + \frac{72}{50} = \frac{87}{50}$
* $(h - \frac{6}{5})^2 = \frac{87}{50}$
* $h - \frac{6}{5} = \pm \sqrt{\frac{87}{50}} = \pm \frac{\sqrt{174}}{10}$
* $h = \frac{12}{10} \pm \frac{\sqrt{174}}{10} = \frac{12 \pm \sqrt{174}}{10}$

9) $10w^2 - 19w - 5 = 0$
* Move constant: $10w^2 - 19w = 5$
* Divide by 10: $w^2 - \frac{19}{10}w = \frac{5}{10} = \frac{1}{2}$
* Half of $-\frac{19}{10}$ is $-\frac{19}{20}$. Square it: $\frac{361}{400}$. Add to both sides.
* Right side: $\frac{1}{2} + \frac{361}{400} = \frac{200}{400} + \frac{361}{400} = \frac{561}{400}$
* $(w - \frac{19}{20})^2 = \frac{561}{400}$
* $w - \frac{19}{20} = \pm \frac{\sqrt{561}}{20}$
* $w = \frac{19 \pm \sqrt{561}}{20}$

10) $m^2 + 13m - 4 = 2$
* Move constant: $m^2 + 13m = 6$
* Half of 13 is $\frac{13}{2}$. Square it: $\frac{169}{4}$. Add to both sides.
* Right side: $6 + \frac{169}{4} = \frac{24}{4} + \frac{169}{4} = \frac{193}{4}$
* $(m + \frac{13}{2})^2 = \frac{193}{4}$
* $m + \frac{13}{2} = \pm \frac{\sqrt{193}}{2}$
* $m = \frac{-13 \pm \sqrt{193}}{2}$

Final Answer:
1) $z = 9 \pm 3\sqrt{5}$
2) $p = -7, -11$
3) $s = 5, \frac{2}{3}$
4) $z = \frac{37 \pm \sqrt{1249}}{12}$
5) $g = \frac{-19 \pm \sqrt{649}}{2}$
6) $x = 1, -2$
7) $w = \frac{12 \pm \sqrt{219}}{3}$
8) $h = \frac{12 \pm \sqrt{174}}{10}$
9) $w = \frac{19 \pm \sqrt{561}}{20}$
10) $m = \frac{-13 \pm \sqrt{193}}{2}$