Solve: (-5+2y)(y-?) = 2y²-9y+10 | Find the Missing Number

Fill in the missing number

$(-5+2y)(y-?)=2y^2-9y+10$

To solve the problem, we'll follow these steps:

• Step 1: Identify and expand the given expression, $(-5+2y)(y-k)$.
• Step 2: Equate it to the provided polynomial $2y^2 - 9y + 10$ and match coefficients.
• Step 3: Solve for $k$.

Let's carry out these steps in detail:

Step 1: Expand $(-5+2y)(y-k)$. Use distributive property:
$(-5+2y)(y-k) = -5(y-k) + 2y(y-k)$.

This results in the following steps:
$-5(y-k) = -5y + 5k$
$2y(y-k) = 2y^2 - 2yk$

Combining these gives:
$-5y + 5k + 2y^2 - 2yk$.

Step 2: Equate this expression to the given polynomial $2y^2 - 9y + 10$:
$2y^2 - 2yk - 5y + 5k = 2y^2 - 9y + 10$.

Step 3: From this, match the coefficients from both the polynomials:

• The coefficient of $y^2$ is already matched as both are 2.
• The coefficient of $y$: $-2k - 5 = -9$.

Solve for $k$:

$-2k - 5 = -9$

$-2k = -9 + 5$

$-2k = -4$

$k = 2$

Therefore, the missing number k is $\mathbf{2}$.