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

$y=3x+10$

Which expression should be added to Y so that :

$y=(x-4)^2$

To solve this problem, we will follow these steps:

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

Let's carry out these steps:

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

$(x - 4)^2 = x^2 - 2 \cdot x \cdot 4 + 4^2$

This simplifies to:

$x^2 - 8x + 16$

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

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

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

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

This simplifies to:

$x^2 - 8x + 16 - 3x - 10 = x^2 - 11x + 6$

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