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1. Write each of the following expressions in the form [tex]\(a(x+b)^2+c\)[/tex], where [tex]\(a, b, c \in \mathbb{Z}\)[/tex].

a. [tex]\(x^2 + 4x\)[/tex]

b. [tex]\(2x^2 - 8x\)[/tex]

c. [tex]\(x^2 + 8x + 7\)[/tex]

Sagot :

GinnyAnswer
Let's rewrite each given expression in the form [tex]\( a(x+b)^2 + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are integers.

### (a) [tex]\( x^2 + 4x \)[/tex]

1. Start with the expression [tex]\( x^2 + 4x \)[/tex].
2. To complete the square, we want to express this as [tex]\( (x + b)^2 + c \)[/tex].
3. We add and subtract the square of half the coefficient of [tex]\( x \)[/tex]:
[tex]\[ x^2 + 4x = x^2 + 4x + 4 - 4 \][/tex]
4. This can be written as:
[tex]\[ (x + 2)^2 - 4 \][/tex]
5. Hence, the expression in the form [tex]\( a(x+b)^2 + c \)[/tex] is:
[tex]\[ (1)(x + 2)^2 - 4 \][/tex]
6. Therefore, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -4 \)[/tex].

### (b) [tex]\( 2x^2 - 8x \)[/tex]

1. Start with the expression [tex]\( 2x^2 - 8x \)[/tex].
2. Factor out the coefficient of [tex]\( x^2 \)[/tex]:
[tex]\[ 2(x^2 - 4x) \][/tex]
3. Now complete the square inside the parenthesis:
[tex]\[ x^2 - 4x = x^2 - 4x + 4 - 4 = (x - 2)^2 - 4 \][/tex]
4. Substituting back gives:
[tex]\[ 2[(x - 2)^2 - 4] = 2(x - 2)^2 - 8 \][/tex]
5. Hence, the expression in the form [tex]\( a(x+b)^2 + c \)[/tex] is:
[tex]\[ (2)(x - 2)^2 - 8 \][/tex]
6. Therefore, [tex]\( a = 2 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -8 \)[/tex].

### (c) [tex]\( x^2 + 8x + 7 \)[/tex]

1. Start with the expression [tex]\( x^2 + 8x + 7 \)[/tex].
2. To complete the square, we write:
[tex]\[ x^2 + 8x + 7 = x^2 + 8x + 16 - 16 + 7 \][/tex]
3. This simplifies to:
[tex]\[ (x + 4)^2 - 16 + 7 \][/tex]
4. Combine the constants:
[tex]\[ (x + 4)^2 - 9 \][/tex]
5. Hence, the expression in the form [tex]\( a(x+b)^2 + c \)[/tex] is:
[tex]\[ (1)(x + 4)^2 - 9 \][/tex]
6. Therefore, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -9 \)[/tex].

### Summary of Results

- (a) [tex]\( x^2 + 4x \)[/tex] can be written as [tex]\( (1)(x + 2)^2 - 4 \)[/tex]
- (b) [tex]\( 2x^2 - 8x \)[/tex] can be written as [tex]\( (2)(x - 2)^2 - 8 \)[/tex]
- (c) [tex]\( x^2 + 8x + 7 \)[/tex] can be written as [tex]\( (1)(x + 4)^2 - 9 \)[/tex]

So, the values of [tex]\( (a, b, c) \)[/tex] for each are:

- (a) [tex]\( (1, 2, -4) \)[/tex]
- (b) [tex]\( (2, -2, -8) \)[/tex]
- (c) [tex]\( (1, 4, -9) \)[/tex]
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