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Completing the square is a method for rewriting a quadratic expression in the form [tex]\(ax^2 + bx + c\)[/tex] into the form [tex]\((x + d)^2 + e\)[/tex], where [tex]\(d\)[/tex] and [tex]\(e\)[/tex] are constants. The steps involved in completing the square are:
1. Start with the quadratic expression [tex]\(ax^2 + bx + c\)[/tex].
2. Move the constant term [tex]\(c\)[/tex] to the other side.
3. Add and subtract [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] to the expression.
4. Factor the perfect square trinomial.
5. Rewrite the expression with the completed square.
Let's apply these steps to the given expressions.
### (a) [tex]\(x^2 - 4x - 5\)[/tex]
1. [tex]\(x^2 - 4x - 5\)[/tex]
2. [tex]\(x^2 - 4x = 5\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-4}{2}\right)^2 = 4\)[/tex]:
[tex]\[ x^2 - 4x + 4 - 4 - 5 = (x - 2)^2 - 9 \][/tex]
4. [tex]\((x - 2)^2 - 9\)[/tex]
### (g) [tex]\(x^2 + 3x - 4\)[/tex]
1. [tex]\(x^2 + 3x - 4\)[/tex]
2. [tex]\(x^2 + 3x = 4\)[/tex]
3. Add and subtract [tex]\(\left(\frac{3}{2}\right)^2 = \frac{9}{4}\)[/tex]:
[tex]\[ x^2 + 3x + \frac{9}{4} - \frac{9}{4} - 4 = \left(x + \frac{3}{2}\right)^2 - \frac{25}{4} \][/tex]
4. [tex]\(\left(x + \frac{3}{2}\right)^2 - \frac{25}{4}\)[/tex]
### (b) [tex]\(x^2 - 2x + 1\)[/tex]
1. [tex]\(x^2 - 2x + 1\)[/tex]
2. [tex]\(x^2 - 2x = -1\)[/tex] (move constant to other side)
3. Add and subtract [tex]\(\left(\frac{-2}{2}\right)^2 = 1\)[/tex]:
[tex]\[ x^2 - 2x + 1 - 1 + 1 = (x - 1)^2 \][/tex]
4. [tex]\((x - 1)^2\)[/tex]
### (h) [tex]\(x^2 - x - 3\)[/tex]
1. [tex]\(x^2 - x - 3\)[/tex]
2. [tex]\(x^2 - x = 3\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-1}{2}\right)^2 = \frac{1}{4}\)[/tex]:
[tex]\[ x^2 - x + \frac{1}{4} - \frac{1}{4} - 3 = \left(x - \frac{1}{2}\right)^2 - \frac{13}{4} \][/tex]
4. [tex]\(\left(x - \frac{1}{2}\right)^2 - \frac{13}{4}\)[/tex]
### (c) [tex]\(x^2 + x + 1\)[/tex]
1. [tex]\(x^2 + x + 1\)[/tex]
2. [tex]\(x^2 + x = -1\)[/tex]
3. Add and subtract [tex]\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)[/tex]:
[tex]\[ x^2 + x + \frac{1}{4} - \frac{1}{4} + 1 = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4} \][/tex]
4. [tex]\(\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}\)[/tex]
### (i) [tex]\(x^2 - 10x + 25\)[/tex]
1. [tex]\(x^2 - 10x + 25\)[/tex]
2. [tex]\(x^2 - 10x = 25\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-10}{2}\right)^2 = 25\)[/tex]:
[tex]\[ x^2 - 10x + 25 - 25 + 25 = (x - 5)^2 \][/tex]
4. [tex]\((x - 5)^2\)[/tex]
### (d) [tex]\(x^2 - 6x + 9\)[/tex]
1. [tex]\(x^2 - 6x + 9\)[/tex]
2. [tex]\(x^2 - 6x = 9\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-6}{2}\right)^2 = 9\)[/tex]:
[tex]\[ x^2 - 6x + 9 - 9 + 9 = (x - 3)^2 \][/tex]
4. [tex]\((x - 3)^2\)[/tex]
### (j) [tex]\(x^2 - 10x\)[/tex]
1. [tex]\(x^2 - 10x\)[/tex]
2. [tex]\(x^2 - 10x = 0\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-10}{2}\right)^2 = 25\)[/tex]:
[tex]\[ x^2 - 10x + 25 - 25 = (x - 5)^2 - 25 \][/tex]
4. [tex]\((x - 5)^2 - 25\)[/tex]
### (e) [tex]\(x^2 - 6x + 7\)[/tex]
1. [tex]\(x^2 - 6x + 7\)[/tex]
2. [tex]\(x^2 - 6x = -7\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-6}{2}\right)^2 = 9\)[/tex]:
[tex]\[ x^2 - 6x + 9 - 9 + 7 = (x - 3)^2 - 2 \][/tex]
4. [tex]\((x - 3)^2 - 2\)[/tex]
### (k) [tex]\(x^2 + 8x + 17\)[/tex]
1. [tex]\(x^2 + 8x + 17\)[/tex]
2. [tex]\(x^2 + 8x = -17\)[/tex]
3. Add and subtract [tex]\(\left(\frac{8}{2}\right)^2 = 16\)[/tex]:
[tex]\[ x^2 + 8x + 16 - 16 + 17 = (x + 4)^2 + 1 \][/tex]
4. [tex]\((x + 4)^2 + 1\)[/tex]
### (f) [tex]\(x^2 - 4x\)[/tex]
1. [tex]\(x^2 - 4x\)[/tex]
2. [tex]\(x^2 - 4x = 0\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-4}{2}\right)^2 = 4\)[/tex]:
[tex]\[ x^2 - 4x + 4 - 4 = (x - 2)^2 - 4 \][/tex]
4. [tex]\((x - 2)^2 - 4\)[/tex]
### (l) [tex]\(x^2 - 12x + 35\)[/tex]
1. [tex]\(x^2 - 12x + 35\)[/tex]
2. [tex]\(x^2 - 12x = -35\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-12}{2}\right)^2 = 36\)[/tex]:
[tex]\[ x^2 - 12x + 36 - 36 + 35 = (x - 6)^2 - 1 \][/tex]
4. [tex]\((x - 6)^2 - 1\)[/tex]
In summary, the completed squares for the given expressions are:
- (a) [tex]\((x - 2)^2 - 9\)[/tex]
- (g) [tex]\(\left(x + \frac{3}{2}\right)^2 - \frac{25}{4}\)[/tex]
- (b) [tex]\((x - 1)^2\)[/tex]
- (h) [tex]\(\left(x - \frac{1}{2}\right)^2 - \frac{13}{4}\)[/tex]
- (c) [tex]\(\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}\)[/tex]
- (i) [tex]\((x - 5)^2\)[/tex]
- (d) [tex]\((x - 3)^2\)[/tex]
- (j) [tex]\((x - 5)^2 - 25\)[/tex]
- (e) [tex]\((x - 3)^2 - 2\)[/tex]
- (k) [tex]\((x + 4)^2 + 1\)[/tex]
- (f) [tex]\((x - 2)^2 - 4\)[/tex]
- (l) [tex]\((x - 6)^2 - 1\)[/tex]
1. Start with the quadratic expression [tex]\(ax^2 + bx + c\)[/tex].
2. Move the constant term [tex]\(c\)[/tex] to the other side.
3. Add and subtract [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] to the expression.
4. Factor the perfect square trinomial.
5. Rewrite the expression with the completed square.
Let's apply these steps to the given expressions.
### (a) [tex]\(x^2 - 4x - 5\)[/tex]
1. [tex]\(x^2 - 4x - 5\)[/tex]
2. [tex]\(x^2 - 4x = 5\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-4}{2}\right)^2 = 4\)[/tex]:
[tex]\[ x^2 - 4x + 4 - 4 - 5 = (x - 2)^2 - 9 \][/tex]
4. [tex]\((x - 2)^2 - 9\)[/tex]
### (g) [tex]\(x^2 + 3x - 4\)[/tex]
1. [tex]\(x^2 + 3x - 4\)[/tex]
2. [tex]\(x^2 + 3x = 4\)[/tex]
3. Add and subtract [tex]\(\left(\frac{3}{2}\right)^2 = \frac{9}{4}\)[/tex]:
[tex]\[ x^2 + 3x + \frac{9}{4} - \frac{9}{4} - 4 = \left(x + \frac{3}{2}\right)^2 - \frac{25}{4} \][/tex]
4. [tex]\(\left(x + \frac{3}{2}\right)^2 - \frac{25}{4}\)[/tex]
### (b) [tex]\(x^2 - 2x + 1\)[/tex]
1. [tex]\(x^2 - 2x + 1\)[/tex]
2. [tex]\(x^2 - 2x = -1\)[/tex] (move constant to other side)
3. Add and subtract [tex]\(\left(\frac{-2}{2}\right)^2 = 1\)[/tex]:
[tex]\[ x^2 - 2x + 1 - 1 + 1 = (x - 1)^2 \][/tex]
4. [tex]\((x - 1)^2\)[/tex]
### (h) [tex]\(x^2 - x - 3\)[/tex]
1. [tex]\(x^2 - x - 3\)[/tex]
2. [tex]\(x^2 - x = 3\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-1}{2}\right)^2 = \frac{1}{4}\)[/tex]:
[tex]\[ x^2 - x + \frac{1}{4} - \frac{1}{4} - 3 = \left(x - \frac{1}{2}\right)^2 - \frac{13}{4} \][/tex]
4. [tex]\(\left(x - \frac{1}{2}\right)^2 - \frac{13}{4}\)[/tex]
### (c) [tex]\(x^2 + x + 1\)[/tex]
1. [tex]\(x^2 + x + 1\)[/tex]
2. [tex]\(x^2 + x = -1\)[/tex]
3. Add and subtract [tex]\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)[/tex]:
[tex]\[ x^2 + x + \frac{1}{4} - \frac{1}{4} + 1 = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4} \][/tex]
4. [tex]\(\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}\)[/tex]
### (i) [tex]\(x^2 - 10x + 25\)[/tex]
1. [tex]\(x^2 - 10x + 25\)[/tex]
2. [tex]\(x^2 - 10x = 25\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-10}{2}\right)^2 = 25\)[/tex]:
[tex]\[ x^2 - 10x + 25 - 25 + 25 = (x - 5)^2 \][/tex]
4. [tex]\((x - 5)^2\)[/tex]
### (d) [tex]\(x^2 - 6x + 9\)[/tex]
1. [tex]\(x^2 - 6x + 9\)[/tex]
2. [tex]\(x^2 - 6x = 9\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-6}{2}\right)^2 = 9\)[/tex]:
[tex]\[ x^2 - 6x + 9 - 9 + 9 = (x - 3)^2 \][/tex]
4. [tex]\((x - 3)^2\)[/tex]
### (j) [tex]\(x^2 - 10x\)[/tex]
1. [tex]\(x^2 - 10x\)[/tex]
2. [tex]\(x^2 - 10x = 0\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-10}{2}\right)^2 = 25\)[/tex]:
[tex]\[ x^2 - 10x + 25 - 25 = (x - 5)^2 - 25 \][/tex]
4. [tex]\((x - 5)^2 - 25\)[/tex]
### (e) [tex]\(x^2 - 6x + 7\)[/tex]
1. [tex]\(x^2 - 6x + 7\)[/tex]
2. [tex]\(x^2 - 6x = -7\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-6}{2}\right)^2 = 9\)[/tex]:
[tex]\[ x^2 - 6x + 9 - 9 + 7 = (x - 3)^2 - 2 \][/tex]
4. [tex]\((x - 3)^2 - 2\)[/tex]
### (k) [tex]\(x^2 + 8x + 17\)[/tex]
1. [tex]\(x^2 + 8x + 17\)[/tex]
2. [tex]\(x^2 + 8x = -17\)[/tex]
3. Add and subtract [tex]\(\left(\frac{8}{2}\right)^2 = 16\)[/tex]:
[tex]\[ x^2 + 8x + 16 - 16 + 17 = (x + 4)^2 + 1 \][/tex]
4. [tex]\((x + 4)^2 + 1\)[/tex]
### (f) [tex]\(x^2 - 4x\)[/tex]
1. [tex]\(x^2 - 4x\)[/tex]
2. [tex]\(x^2 - 4x = 0\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-4}{2}\right)^2 = 4\)[/tex]:
[tex]\[ x^2 - 4x + 4 - 4 = (x - 2)^2 - 4 \][/tex]
4. [tex]\((x - 2)^2 - 4\)[/tex]
### (l) [tex]\(x^2 - 12x + 35\)[/tex]
1. [tex]\(x^2 - 12x + 35\)[/tex]
2. [tex]\(x^2 - 12x = -35\)[/tex]
3. Add and subtract [tex]\(\left(\frac{-12}{2}\right)^2 = 36\)[/tex]:
[tex]\[ x^2 - 12x + 36 - 36 + 35 = (x - 6)^2 - 1 \][/tex]
4. [tex]\((x - 6)^2 - 1\)[/tex]
In summary, the completed squares for the given expressions are:
- (a) [tex]\((x - 2)^2 - 9\)[/tex]
- (g) [tex]\(\left(x + \frac{3}{2}\right)^2 - \frac{25}{4}\)[/tex]
- (b) [tex]\((x - 1)^2\)[/tex]
- (h) [tex]\(\left(x - \frac{1}{2}\right)^2 - \frac{13}{4}\)[/tex]
- (c) [tex]\(\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}\)[/tex]
- (i) [tex]\((x - 5)^2\)[/tex]
- (d) [tex]\((x - 3)^2\)[/tex]
- (j) [tex]\((x - 5)^2 - 25\)[/tex]
- (e) [tex]\((x - 3)^2 - 2\)[/tex]
- (k) [tex]\((x + 4)^2 + 1\)[/tex]
- (f) [tex]\((x - 2)^2 - 4\)[/tex]
- (l) [tex]\((x - 6)^2 - 1\)[/tex]
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