Completing the Square practice worksheet from Corbett Maths, including examples, workout problems, and a video link.

Let's solve Question 1 and Question 2 step by step using the method of completing the square.

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🔷 Question 1: Write the following expressions in the form $(x + a)^2 + b$

We use the identity:
$$
x^2 + bx = (x + \frac{b}{2})^2 - \left(\frac{b}{2}\right)^2
$$

So, for each expression $x^2 + bx + c$, we rewrite it as:
$$
(x + \frac{b}{2})^2 - \left(\frac{b}{2}\right)^2 + c
$$

Let’s go through each part:

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(a) $x^2 + 8x + 1$
- Half of 8 is 4 → $(x + 4)^2$
- $(4)^2 = 16$
- So: $(x + 4)^2 - 16 + 1 = (x + 4)^2 - 15$

✔ Answer: $\boxed{(x + 4)^2 - 15}$

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(b) $x^2 + 10x + 3$
- Half of 10 is 5 → $(x + 5)^2$
- $5^2 = 25$
- $(x + 5)^2 - 25 + 3 = (x + 5)^2 - 22$

✔ Answer: $\boxed{(x + 5)^2 - 22}$

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(c) $x^2 + 2x - 1$
- Half of 2 is 1 → $(x + 1)^2$
- $1^2 = 1$
- $(x + 1)^2 - 1 - 1 = (x + 1)^2 - 2$

✔ Answer: $\boxed{(x + 1)^2 - 2}$

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(d) $x^2 - 6x - 10$
- Half of -6 is -3 → $(x - 3)^2$
- $(-3)^2 = 9$
- $(x - 3)^2 - 9 - 10 = (x - 3)^2 - 19$

✔ Answer: $\boxed{(x - 3)^2 - 19}$

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(e) $x^2 - 4x - 13$
- Half of -4 is -2 → $(x - 2)^2$
- $(-2)^2 = 4$
- $(x - 2)^2 - 4 - 13 = (x - 2)^2 - 17$

✔ Answer: $\boxed{(x - 2)^2 - 17}$

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(f) $x^2 - 12x + 3$
- Half of -12 is -6 → $(x - 6)^2$
- $(-6)^2 = 36$
- $(x - 6)^2 - 36 + 3 = (x - 6)^2 - 33$

✔ Answer: $\boxed{(x - 6)^2 - 33}$

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(g) $x^2 + 14x + 3$
- Half of 14 is 7 → $(x + 7)^2$
- $7^2 = 49$
- $(x + 7)^2 - 49 + 3 = (x + 7)^2 - 46$

✔ Answer: $\boxed{(x + 7)^2 - 46}$

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(h) $x^2 - 2x - 15$
- Half of -2 is -1 → $(x - 1)^2$
- $(-1)^2 = 1$
- $(x - 1)^2 - 1 - 15 = (x - 1)^2 - 16$

✔ Answer: $\boxed{(x - 1)^2 - 16}$

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(i) $x^2 + 4x - 11$
- Half of 4 is 2 → $(x + 2)^2$
- $2^2 = 4$
- $(x + 2)^2 - 4 - 11 = (x + 2)^2 - 15$

✔ Answer: $\boxed{(x + 2)^2 - 15}$

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(j) $x^2 + x - 8$
- Half of 1 is $0.5$ → $(x + 0.5)^2$
- $(0.5)^2 = 0.25$
- $(x + 0.5)^2 - 0.25 - 8 = (x + 0.5)^2 - 8.25$

Or write as fractions:
$(x + \frac{1}{2})^2 - \frac{1}{4} - 8 = (x + \frac{1}{2})^2 - \frac{33}{4}$

✔ Answer: $\boxed{\left(x + \frac{1}{2}\right)^2 - \frac{33}{4}}$

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(k) $x^2 + 3x + 1$
- Half of 3 is $1.5 = \frac{3}{2}$ → $(x + \frac{3}{2})^2$
- $(\frac{3}{2})^2 = \frac{9}{4}$
- $(x + \frac{3}{2})^2 - \frac{9}{4} + 1 = (x + \frac{3}{2})^2 - \frac{5}{4}$

✔ Answer: $\boxed{\left(x + \frac{3}{2}\right)^2 - \frac{5}{4}}$

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(l) $x^2 - 7x - 2$
- Half of -7 is $-\frac{7}{2}$ → $(x - \frac{7}{2})^2$
- $(-\frac{7}{2})^2 = \frac{49}{4}$
- $(x - \frac{7}{2})^2 - \frac{49}{4} - 2 = (x - \frac{7}{2})^2 - \frac{57}{4}$

✔ Answer: $\boxed{\left(x - \frac{7}{2}\right)^2 - \frac{57}{4}}$

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(m) $x^2 - 9x - 1$
- Half of -9 is $-\frac{9}{2}$ → $(x - \frac{9}{2})^2$
- $(-\frac{9}{2})^2 = \frac{81}{4}$
- $(x - \frac{9}{2})^2 - \frac{81}{4} - 1 = (x - \frac{9}{2})^2 - \frac{85}{4}$

✔ Answer: $\boxed{\left(x - \frac{9}{2}\right)^2 - \frac{85}{4}}$

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(n) $x^2 + 11x + 3$
- Half of 11 is $\frac{11}{2}$ → $(x + \frac{11}{2})^2$
- $(\frac{11}{2})^2 = \frac{121}{4}$
- $(x + \frac{11}{2})^2 - \frac{121}{4} + 3 = (x + \frac{11}{2})^2 - \frac{109}{4}$

✔ Answer: $\boxed{\left(x + \frac{11}{2}\right)^2 - \frac{109}{4}}$

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(o) $x^2 - 100x - 25$
- Half of -100 is -50 → $(x - 50)^2$
- $(-50)^2 = 2500$
- $(x - 50)^2 - 2500 - 25 = (x - 50)^2 - 2525$

✔ Answer: $\boxed{(x - 50)^2 - 2525}$

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## ✔ Question 1 Summary:

| Expression | Completed Square Form |
|----------|------------------------|
| (a) $x^2 + 8x + 1$ | $(x + 4)^2 - 15$ |
| (b) $x^2 + 10x + 3$ | $(x + 5)^2 - 22$ |
| (c) $x^2 + 2x - 1$ | $(x + 1)^2 - 2$ |
| (d) $x^2 - 6x - 10$ | $(x - 3)^2 - 19$ |
| (e) $x^2 - 4x - 13$ | $(x - 2)^2 - 17$ |
| (f) $x^2 - 12x + 3$ | $(x - 6)^2 - 33$ |
| (g) $x^2 + 14x + 3$ | $(x + 7)^2 - 46$ |
| (h) $x^2 - 2x - 15$ | $(x - 1)^2 - 16$ |
| (i) $x^2 + 4x - 11$ | $(x + 2)^2 - 15$ |
| (j) $x^2 + x - 8$ | $\left(x + \frac{1}{2}\right)^2 - \frac{33}{4}$ |
| (k) $x^2 + 3x + 1$ | $\left(x + \frac{3}{2}\right)^2 - \frac{5}{4}$ |
| (l) $x^2 - 7x - 2$ | $\left(x - \frac{7}{2}\right)^2 - \frac{57}{4}$ |
| (m) $x^2 - 9x - 1$ | $\left(x - \frac{9}{2}\right)^2 - \frac{85}{4}$ |
| (n) $x^2 + 11x + 3$ | $\left(x + \frac{11}{2}\right)^2 - \frac{109}{4}$ |
| (o) $x^2 - 100x - 25$ | $(x - 50)^2 - 2525$ |

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## 🔷 Question 2: Solve the equations using completing the square

We’ll solve each equation by rewriting in the form $(x + a)^2 = k$, then take square roots.

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(a) $x^2 + 4x + 1 = 0$
Move constant: $x^2 + 4x = -1$
Complete the square: half of 4 is 2 → add $2^2 = 4$ to both sides
$x^2 + 4x + 4 = -1 + 4 = 3$
$(x + 2)^2 = 3$
Take square root: $x + 2 = \pm\sqrt{3}$
$x = -2 \pm \sqrt{3}$

✔ Answer: $\boxed{x = -2 \pm \sqrt{3}}$

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(b) $x^2 + 8x - 10 = 0$
$x^2 + 8x = 10$
Half of 8 is 4 → add $16$:
$x^2 + 8x + 16 = 10 + 16 = 26$
$(x + 4)^2 = 26$
$x + 4 = \pm\sqrt{26}$
$x = -4 \pm \sqrt{26}$

✔ Answer: $\boxed{x = -4 \pm \sqrt{26}}$

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(c) $x^2 + 14x - 4 = 0$
$x^2 + 14x = 4$
Half of 14 is 7 → add $49$:
$x^2 + 14x + 49 = 4 + 49 = 53$
$(x + 7)^2 = 53$
$x + 7 = \pm\sqrt{53}$
$x = -7 \pm \sqrt{53}$

✔ Answer: $\boxed{x = -7 \pm \sqrt{53}}$

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(d) $x^2 - 8x - 2 = 0$
$x^2 - 8x = 2$
Half of -8 is -4 → add $16$:
$x^2 - 8x + 16 = 2 + 16 = 18$
$(x - 4)^2 = 18$
$x - 4 = \pm\sqrt{18} = \pm 3\sqrt{2}$
$x = 4 \pm 3\sqrt{2}$

✔ Answer: $\boxed{x = 4 \pm 3\sqrt{2}}$

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(e) $x^2 - 10x + 10 = 0$
$x^2 - 10x = -10$
Half of -10 is -5 → add $25$:
$x^2 - 10x + 25 = -10 + 25 = 15$
$(x - 5)^2 = 15$
$x - 5 = \pm\sqrt{15}$
$x = 5 \pm \sqrt{15}$

✔ Answer: $\boxed{x = 5 \pm \sqrt{15}}$

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(f) $x^2 + 18x + 7 = 0$
$x^2 + 18x = -7$
Half of 18 is 9 → add $81$:
$x^2 + 18x + 81 = -7 + 81 = 74$
$(x + 9)^2 = 74$
$x + 9 = \pm\sqrt{74}$
$x = -9 \pm \sqrt{74}$

✔ Answer: $\boxed{x = -9 \pm \sqrt{74}}$

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## ✔ Question 2 Summary:

| Equation | Solution |
|--------|---------|
| (a) $x^2 + 4x + 1 = 0$ | $x = -2 \pm \sqrt{3}$ |
| (b) $x^2 + 8x - 10 = 0$ | $x = -4 \pm \sqrt{26}$ |
| (c) $x^2 + 14x - 4 = 0$ | $x = -7 \pm \sqrt{53}$ |
| (d) $x^2 - 8x - 2 = 0$ | $x = 4 \pm 3\sqrt{2}$ |
| (e) $x^2 - 10x + 10 = 0$ | $x = 5 \pm \sqrt{15}$ |
| (f) $x^2 + 18x + 7 = 0$ | $x = -9 \pm \sqrt{74}$ |

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🧠 Key Steps Recap:

1. Move constant term to other side.
2. Take half of coefficient of $x$, square it, add to both sides.
3. Write left side as perfect square.
4. Take square root of both sides.
5. Solve for $x$.

Let me know if you'd like these solutions graphed or checked with the quadratic formula!