Solving Quadratic Equations by Completing the Squares Worksheets - Free Printable

Let's solve each quadratic equation by completing the square. I'll go through each problem step-by-step.

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1) $ 6k^2 + 21 = -17k $

Step 1: Move all terms to one side:
$$
6k^2 + 17k + 21 = 0
$$

Step 2: Divide by the coefficient of $ k^2 $ (which is 6):
$$
k^2 + \frac{17}{6}k + \frac{21}{6} = 0 \Rightarrow k^2 + \frac{17}{6}k + \frac{7}{2} = 0
$$

Step 3: Move constant to the other side:
$$
k^2 + \frac{17}{6}k = -\frac{7}{2}
$$

Step 4: Complete the square:
Take half of $ \frac{17}{6} $: $ \frac{17}{12} $, square it: $ \left(\frac{17}{12}\right)^2 = \frac{289}{144} $

Add to both sides:
$$
k^2 + \frac{17}{6}k + \frac{289}{144} = -\frac{7}{2} + \frac{289}{144}
$$

Convert $ -\frac{7}{2} $ to 144 denominator:
$$
-\frac{7}{2} = -\frac{504}{144}
\Rightarrow -\frac{504}{144} + \frac{289}{144} = -\frac{215}{144}
$$

So:
$$
\left(k + \frac{17}{12}\right)^2 = -\frac{215}{144}
$$

Step 5: Take square root:
$$
k + \frac{17}{12} = \pm \sqrt{-\frac{215}{144}} = \pm \frac{\sqrt{215}}{12}i
$$

$$
k = -\frac{17}{12} \pm \frac{\sqrt{215}}{12}i
$$

✔ Answer: $ k = -\frac{17}{12} \pm \frac{\sqrt{215}}{12}i $

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2) $ 3m^2 - 1 = -10m $

Step 1: Bring all terms to one side:
$$
3m^2 + 10m - 1 = 0
$$

Step 2: Divide by 3:
$$
m^2 + \frac{10}{3}m - \frac{1}{3} = 0
$$

Step 3: Move constant:
$$
m^2 + \frac{10}{3}m = \frac{1}{3}
$$

Step 4: Half of $ \frac{10}{3} $ is $ \frac{5}{3} $, square: $ \frac{25}{9} $

Add to both sides:
$$
m^2 + \frac{10}{3}m + \frac{25}{9} = \frac{1}{3} + \frac{25}{9} = \frac{3}{9} + \frac{25}{9} = \frac{28}{9}
$$

$$
\left(m + \frac{5}{3}\right)^2 = \frac{28}{9}
$$

Step 5: Square root:
$$
m + \frac{5}{3} = \pm \frac{\sqrt{28}}{3} = \pm \frac{2\sqrt{7}}{3}
$$

$$
m = -\frac{5}{3} \pm \frac{2\sqrt{7}}{3}
$$

✔ Answer: $ m = \frac{-5 \pm 2\sqrt{7}}{3} $

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3) $ \frac{1}{2}p^2 + p + 6 = 0 $

Step 1: Multiply entire equation by 2 to eliminate fraction:
$$
p^2 + 2p + 12 = 0
$$

Step 2: Move constant:
$$
p^2 + 2p = -12
$$

Step 3: Half of 2 is 1, square is 1:
$$
p^2 + 2p + 1 = -12 + 1 = -11
$$

$$
(p + 1)^2 = -11
$$

Step 4: Square root:
$$
p + 1 = \pm \sqrt{-11} = \pm i\sqrt{11}
\Rightarrow p = -1 \pm i\sqrt{11}
$$

✔ Answer: $ p = -1 \pm i\sqrt{11} $

---

4) $ \frac{1}{4}t^2 + \frac{1}{2}t - \frac{1}{16} = 0 $

Step 1: Multiply by 16 to eliminate denominators:
$$
4t^2 + 8t - 1 = 0
$$

Step 2: Divide by 4:
$$
t^2 + 2t - \frac{1}{4} = 0
$$

Step 3: Move constant:
$$
t^2 + 2t = \frac{1}{4}
$$

Step 4: Half of 2 is 1, square is 1:
$$
t^2 + 2t + 1 = \frac{1}{4} + 1 = \frac{5}{4}
\Rightarrow (t + 1)^2 = \frac{5}{4}
$$

Step 5: Square root:
$$
t + 1 = \pm \frac{\sqrt{5}}{2}
\Rightarrow t = -1 \pm \frac{\sqrt{5}}{2}
$$

✔ Answer: $ t = -1 \pm \frac{\sqrt{5}}{2} $

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5) $ 4u^2 - 3u - \frac{1}{2} = 0 $

Step 1: Multiply by 2:
$$
8u^2 - 6u - 1 = 0
$$

Divide by 8:
$$
u^2 - \frac{3}{4}u - \frac{1}{8} = 0
$$

Move constant:
$$
u^2 - \frac{3}{4}u = \frac{1}{8}
$$

Half of $ -\frac{3}{4} $ is $ -\frac{3}{8} $, square: $ \frac{9}{64} $

Add to both sides:
$$
u^2 - \frac{3}{4}u + \frac{9}{64} = \frac{1}{8} + \frac{9}{64} = \frac{8}{64} + \frac{9}{64} = \frac{17}{64}
$$

$$
\left(u - \frac{3}{8}\right)^2 = \frac{17}{64}
$$

Square root:
$$
u - \frac{3}{8} = \pm \frac{\sqrt{17}}{8}
\Rightarrow u = \frac{3}{8} \pm \frac{\sqrt{17}}{8}
$$

✔ Answer: $ u = \frac{3 \pm \sqrt{17}}{8} $

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6) $ 9w^2 + 4w - 8 = 0 $

Step 1: Divide by 9:
$$
w^2 + \frac{4}{9}w - \frac{8}{9} = 0
$$

Move constant:
$$
w^2 + \frac{4}{9}w = \frac{8}{9}
$$

Half of $ \frac{4}{9} $ is $ \frac{2}{9} $, square: $ \frac{4}{81} $

Add to both sides:
$$
w^2 + \frac{4}{9}w + \frac{4}{81} = \frac{8}{9} + \frac{4}{81} = \frac{72}{81} + \frac{4}{81} = \frac{76}{81}
$$

$$
\left(w + \frac{2}{9}\right)^2 = \frac{76}{81}
$$

Square root:
$$
w + \frac{2}{9} = \pm \frac{\sqrt{76}}{9} = \pm \frac{2\sqrt{19}}{9}
\Rightarrow w = -\frac{2}{9} \pm \frac{2\sqrt{19}}{9}
$$

✔ Answer: $ w = \frac{-2 \pm 2\sqrt{19}}{9} $

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7) $ 5z^2 + 8 = -34z $

Step 1: Move all terms:
$$
5z^2 + 34z + 8 = 0
$$

Divide by 5:
$$
z^2 + \frac{34}{5}z + \frac{8}{5} = 0
$$

Move constant:
$$
z^2 + \frac{34}{5}z = -\frac{8}{5}
$$

Half of $ \frac{34}{5} $ is $ \frac{17}{5} $, square: $ \frac{289}{25} $

Add to both sides:
$$
z^2 + \frac{34}{5}z + \frac{289}{25} = -\frac{8}{5} + \frac{289}{25} = -\frac{40}{25} + \frac{289}{25} = \frac{249}{25}
$$

$$
\left(z + \frac{17}{5}\right)^2 = \frac{249}{25}
$$

Square root:
$$
z + \frac{17}{5} = \pm \frac{\sqrt{249}}{5}
\Rightarrow z = -\frac{17}{5} \pm \frac{\sqrt{249}}{5}
$$

✔ Answer: $ z = \frac{-17 \pm \sqrt{249}}{5} $

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8) $ \frac{2}{3}x^2 + \frac{2}{3}x - 2 = 0 $

Step 1: Multiply by 3:
$$
2x^2 + 2x - 6 = 0
$$

Divide by 2:
$$
x^2 + x - 3 = 0
$$

Move constant:
$$
x^2 + x = 3
$$

Half of 1 is $ \frac{1}{2} $, square: $ \frac{1}{4} $

Add:
$$
x^2 + x + \frac{1}{4} = 3 + \frac{1}{4} = \frac{13}{4}
\Rightarrow \left(x + \frac{1}{2}\right)^2 = \frac{13}{4}
$$

Square root:
$$
x + \frac{1}{2} = \pm \frac{\sqrt{13}}{2}
\Rightarrow x = -\frac{1}{2} \pm \frac{\sqrt{13}}{2}
$$

✔ Answer: $ x = \frac{-1 \pm \sqrt{13}}{2} $

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✔ Final Answers:

1) $ k = -\frac{17}{12} \pm \frac{\sqrt{215}}{12}i $
2) $ m = \frac{-5 \pm 2\sqrt{7}}{3} $
3) $ p = -1 \pm i\sqrt{11} $
4) $ t = -1 \pm \frac{\sqrt{5}}{2} $
5) $ u = \frac{3 \pm \sqrt{17}}{8} $
6) $ w = \frac{-2 \pm 2\sqrt{19}}{9} $
7) $ z = \frac{-17 \pm \sqrt{249}}{5} $
8) $ x = \frac{-1 \pm \sqrt{13}}{2} $

Let me know if you'd like these in decimal form or need help graphing!