Transform y=3x+10 into (x-4)²: Finding the Missing Expression

Question

y=3x+10 y=3x+10

Which expression should be added to Y so that :

y=(x4)2 y=(x-4)^2

Video Solution

Solution Steps

00:00 Find the correct expression so that the equation is satisfied
00:05 Use the short multiplication formulas to expand the brackets
00:16 Find the difference between the expressions
00:29 Add the differences to the expression
00:32 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we will follow these steps:

  • Step 1: Expand the expression (x4)2 (x - 4)^2 .
  • Step 2: Compare the expanded quadratic expression with 3x+10 3x + 10 .
  • Step 3: Determine the additional expression needed.

Let's carry out these steps:

Step 1: First, expand (x4)2 (x - 4)^2 . Using the formula for the square of a binomial, we have:

(x4)2=x22x4+42 (x - 4)^2 = x^2 - 2 \cdot x \cdot 4 + 4^2

This simplifies to:

x28x+16 x^2 - 8x + 16

Step 2: We need to convert y=3x+10 y = 3x + 10 into this expanded form. That means 3x+10 3x + 10 should become x28x+16 x^2 - 8x + 16 .

Step 3: The expression to add is the difference between x28x+16 x^2 - 8x + 16 and 3x+10 3x + 10 :

Subtract 3x+10 3x + 10 from x28x+16 x^2 - 8x + 16 :

(x28x+16)(3x+10) (x^2 - 8x + 16) - (3x + 10)

This simplifies to:

x28x+163x10=x211x+6 x^2 - 8x + 16 - 3x - 10 = x^2 - 11x + 6

Therefore, the expression that should be added to y y is x211x+6 x^2 - 11x + 6 .

Answer

x211x+6 x^2-11x+6