Math  /  Algebra

QuestionQUESTION 6 / 10 - LEVEL 2 10 MARK(S) 2x(x+3)2 x(x+3) can be expressed in the form of a(x+1)2+b(x+1)+ca(x+1)^{2}+b(x+1)+c. What is the value of bb ?

Studdy Solution

STEP 1

1. We need to express the quadratic expression 2x(x+3) 2x(x+3) in the form a(x+1)2+b(x+1)+c a(x+1)^2 + b(x+1) + c .
2. The expression 2x(x+3) 2x(x+3) is a quadratic polynomial.
3. The goal is to find the coefficient b b in the transformed expression.

STEP 2

1. Expand the given expression 2x(x+3) 2x(x+3) .
2. Expand the target form a(x+1)2+b(x+1)+c a(x+1)^2 + b(x+1) + c .
3. Equate the coefficients of like terms.
4. Solve for the coefficient b b .

STEP 3

Expand the expression 2x(x+3) 2x(x+3) :
2x(x+3)=2x2+6x 2x(x+3) = 2x^2 + 6x

STEP 4

Expand the expression a(x+1)2+b(x+1)+c a(x+1)^2 + b(x+1) + c :
a(x+1)2=a(x2+2x+1)=ax2+2ax+a a(x+1)^2 = a(x^2 + 2x + 1) = ax^2 + 2ax + a b(x+1)=bx+b b(x+1) = bx + b
Combine these:
ax2+2ax+a+bx+b+c ax^2 + 2ax + a + bx + b + c

STEP 5

Combine like terms in the expanded form:
ax2+(2a+b)x+(a+b+c) ax^2 + (2a + b)x + (a + b + c)

STEP 6

Equate the coefficients of the expanded forms 2x2+6x 2x^2 + 6x and ax2+(2a+b)x+(a+b+c) ax^2 + (2a + b)x + (a + b + c) :
1. ax2 ax^2 term: a=2 a = 2
2. (2a+b)x (2a + b)x term: 2a+b=6 2a + b = 6
3. Constant term: a+b+c=0 a + b + c = 0

STEP 7

Solve for b b using the equations from STEP_4:
1. From a=2 a = 2 , substitute into 2a+b=6 2a + b = 6 :
2(2)+b=6 2(2) + b = 6 4+b=6 4 + b = 6 b=2 b = 2
The value of b b is:
2 \boxed{2}

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