Question

the diagram representing the factorization of ( x^2 - 9x + 18 ) is partially completed.
diagram with cells: top row ( x ), ( -3 ); middle left ( x ), middle ( x^2 ), middle right (? ); bottom left ( -6 ), bottom middle (? ), bottom right ( 18 )
which two terms are missing from the diagram?

• ( -x ) and ( -8x )
• ( -2x ) and ( -7x )
• ( -3x ) and ( -6x )
• ( -4x ) and ( -5x )

Explanation:

Step1: Recall factoring via box method

For factoring \(x^2 - 9x + 18\), the box method has four cells in the middle row and column. The top - left is \(x^2\) (from \(x\times x\)), bottom - right is \(18\) (from \(- 3\times - 6\)). The middle terms come from multiplying the row and column factors. The first row is \(x\) and \(-3\), the first column is \(x\) and \(-6\).

Step2: Calculate the middle cells

The cell at the intersection of the first row (\(x\)) and second column (\(-6\)): \(x\times(-6)=-6x\).
The cell at the intersection of the second row (\(-6\)) and second column (\(-3\)): \((-6)\times(-3) = 18\) (already given). Wait, the middle cell (row 2, column 2) is the sum of the two middle terms. Wait, actually, the quadratic is \(x^2-9x + 18=(x - 3)(x - 6)=x^2-6x-3x + 18\). So the two middle terms in the box (the non - corner middle cells) should be \(-3x\) (from \(x\times(-3)\)) and \(-6x\) (from \(x\times(-6)\))? Wait, no, let's re - examine the box. The top row: \(x\) and \(-3\), left column: \(x\) and \(-6\). So the cell (row 2, column 2) is \((-6)\times(-3) = 18\), cell (row 1, column 2) is \(x\times(-3)=-3x\), cell (row 2, column 1) is \(x\times(-6)=-6x\), and cell (row 1, column 1) is \(x\times x=x^2\). Then the middle row (the sum of the two middle cells) should be \(-3x-6x=-9x\), which matches the quadratic \(x^2-9x + 18\). So the missing terms are \(-3x\) and \(-6x\).

Answer:

-3x and -6x (the option with -3x and -6x)