Completing the Square worksheet

1. Completed-square form: 2(x − 3.5)² − 12.5

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $2\times x^2+\mathrm{-14}\times x+12$$=2\text{(}{x}^2+\mathrm{-7}\times x\text{)}+12$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-14}}{2\times 2}$$=\mathrm{-3.5}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-3.5}\right)}^2$$=12.25$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=12-2\times 12.25$$=\mathrm{-12.5}$

5. So in completed-square (vertex) form, the expression is

   $\text{2(x − 3.5)² − 12.5}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{12.5}{2}$$=6.25$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=3.5\pm \sqrt{6.25}$$=3.5\pm 2.5$

8. The two solutions are

   $\text{x = 6 or x = 1}$

2. Completed-square form: 2(x − 2.5)² − 5.5

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $2\times x^2+\mathrm{-10}\times x+7$$=2\text{(}{x}^2+\mathrm{-5}\times x\text{)}+7$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-10}}{2\times 2}$$=\mathrm{-2.5}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-2.5}\right)}^2$$=6.25$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=7-2\times 6.25$$=\mathrm{-5.5}$

5. So in completed-square (vertex) form, the expression is

   $\text{2(x − 2.5)² − 5.5}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{5.5}{2}$$=2.75$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=2.5\pm \sqrt{2.75}$$=2.5\pm 1.658312$

8. The two solutions are

   $\text{x = 4.158312 or x = 0.841688}$

3. Completed-square form: 2(x − 2.25)² − 2.125

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $2\times x^2+\mathrm{-9}\times x+8$$=2\text{(}{x}^2+\mathrm{-4.5}\times x\text{)}+8$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-9}}{2\times 2}$$=\mathrm{-2.25}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-2.25}\right)}^2$$=5.0625$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=8-2\times 5.0625$$=\mathrm{-2.125}$

5. So in completed-square (vertex) form, the expression is

   $\text{2(x − 2.25)² − 2.125}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{2.125}{2}$$=1.0625$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=2.25\pm \sqrt{1.0625}$$=2.25\pm 1.030776$

8. The two solutions are

   $\text{x = 3.280776 or x = 1.219224}$

4. Completed-square form: 2(x − 3)² − 4

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $2\times x^2+\mathrm{-12}\times x+14$$=2\text{(}{x}^2+\mathrm{-6}\times x\text{)}+14$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-12}}{2\times 2}$$=\mathrm{-3}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-3}\right)}^2$$=9$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=14-2\times 9$$=\mathrm{-4}$

5. So in completed-square (vertex) form, the expression is

   $\text{2(x − 3)² − 4}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{4}{2}$$=2$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=3\pm \sqrt{2}$$=3\pm 1.414214$

8. The two solutions are

   $\text{x = 4.414214 or x = 1.585786}$

5. Completed-square form: 3(x − 2.5)² − 7.75

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $3\times x^2+\mathrm{-15}\times x+11$$=3\text{(}{x}^2+\mathrm{-5}\times x\text{)}+11$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-15}}{2\times 3}$$=\mathrm{-2.5}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-2.5}\right)}^2$$=6.25$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=11-3\times 6.25$$=\mathrm{-7.75}$

5. So in completed-square (vertex) form, the expression is

   $\text{3(x − 2.5)² − 7.75}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{7.75}{3}$$=2.583333$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=2.5\pm \sqrt{2.583333}$$=2.5\pm 1.607275$

8. The two solutions are

   $\text{x = 4.107275 or x = 0.892725}$

6. Completed-square form: 3(x − 1.666667)² + 2.666667

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $3\times x^2+\mathrm{-10}\times x+11$$=3\text{(}{x}^2+\mathrm{-3.333333}\times x\text{)}+11$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-10}}{2\times 3}$$=\mathrm{-1.666667}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-1.666667}\right)}^2$$=2.777778$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=11-3\times 2.777778$$=2.666667$

5. So in completed-square (vertex) form, the expression is

   $\text{3(x − 1.666667)² + 2.666667}$

6. Getting the square on its own gives a negative number, (x − h)² = −k ÷ a. No real number squares to a negative, so there are no real solutions

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{\mathrm{-2.666667}}{3}$$=\mathrm{-0.888889}$

7. Its square roots use i, the square root of −1: ±i√|−k ÷ a|. That gives a complex pair

   $\text{x = 1.666667 ± 0.942809i}$

7. Completed-square form: 3(x − 2)² − 2

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $3\times x^2+\mathrm{-12}\times x+10$$=3\text{(}{x}^2+\mathrm{-4}\times x\text{)}+10$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-12}}{2\times 3}$$=\mathrm{-2}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-2}\right)}^2$$=4$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=10-3\times 4$$=\mathrm{-2}$

5. So in completed-square (vertex) form, the expression is

   $\text{3(x − 2)² − 2}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{2}{3}$$=0.666667$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=2\pm \sqrt{0.666667}$$=2\pm 0.816497$

8. The two solutions are

   $\text{x = 2.816497 or x = 1.183503}$

8. Completed-square form: (x − 5.5)² − 21.25

1. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-11}}{2\times 1}$$=\mathrm{-5.5}$

2. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-5.5}\right)}^2$$=30.25$

3. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=9-1\times 30.25$$=\mathrm{-21.25}$

4. So in completed-square (vertex) form, the expression is

   $\text{(x − 5.5)² − 21.25}$

5. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{21.25}{1}$$=21.25$

6. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=5.5\pm \sqrt{21.25}$$=5.5\pm 4.609772$

7. The two solutions are

   $\text{x = 10.109772 or x = 0.890228}$

9. Completed-square form: 2(x − 2.75)² − 6.125

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $2\times x^2+\mathrm{-11}\times x+9$$=2\text{(}{x}^2+\mathrm{-5.5}\times x\text{)}+9$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-11}}{2\times 2}$$=\mathrm{-2.75}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-2.75}\right)}^2$$=7.5625$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=9-2\times 7.5625$$=\mathrm{-6.125}$

5. So in completed-square (vertex) form, the expression is

   $\text{2(x − 2.75)² − 6.125}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{6.125}{2}$$=3.0625$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=2.75\pm \sqrt{3.0625}$$=2.75\pm 1.75$

8. The two solutions are

   $\text{x = 4.5 or x = 1}$

10. Completed-square form: 2(x − 2.5)² − 0.5

1. Factor a out of the x² and x terms, so the bracket starts with a plain x²

   $2\times x^2+\mathrm{-10}\times x+12$$=2\text{(}{x}^2+\mathrm{-5}\times x\text{)}+12$

2. Halve the number in front of x inside the bracket. This is the number d that goes inside the square

   $d$$=\frac{\mathrm{-10}}{2\times 2}$$=\mathrm{-2.5}$

3. Because (x + d)² = x² + 2dx + d², adding d² and then taking it away again completes the square without changing the value

   $d^2$$={\left(\mathrm{-2.5}\right)}^2$$=6.25$

4. Multiply the d² you took away by a, and combine it with c. This gives the plain number k

   $k$$=12-2\times 6.25$$=\mathrm{-0.5}$

5. So in completed-square (vertex) form, the expression is

   $\text{2(x − 2.5)² − 0.5}$

6. To solve the equation = 0, get the square on its own: (x − h)² = −k ÷ a

   ${\mathrm{(x\; -\; h)}}^2$$=\frac{0.5}{2}$$=0.25$

7. Take the square root of both sides. Remember ±, since a positive or negative number squares to the same value. Then add h

   $x$$=2.5\pm \sqrt{0.25}$$=2.5\pm 0.5$

8. The two solutions are

   $\text{x = 3 or x = 2}$