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Mathematics

(i) What should be added to 7a - 9b + 13c to get 9a + b - c?

(ii) What should be added to 1 + 2x - 3x2 to get x2 - x - 1?

(iii) What should be added to m2 - 2mn + 5n2 to get n2 + mn - m2?

Algebraic Expressions

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Answer

(i) What should be added to 7a - 9b + 13c to get 9a + b - c?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract 7a - 9b + 13c from 9a + b - c:

9a+bc+7a9b+13c7a+2a+10b14c\begin{array}{rcccc} 9a & + & b & - & c \\ +7a & - & 9b & + & 13c \\ -\phantom{7a} & + & & - \\ \hline 2a & + & 10b & - & 14c \\ \hline \end{array}

∴ The required expression is 2a + 10b - 14c

(ii) What should be added to 1 + 2x - 3x2 to get x2 - x - 1?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract 1 + 2x - 3x2 from x2 - x - 1:

Arranging the expressions in descending powers of x (x2, x, constant).

x2x13x2+2x+1+3x24x23x2\begin{array}{rcccc} x^2 & - & x & - & 1 \\ -3x^2 & + & 2x & + & 1 \\ +\phantom{3x^2} & - & & - \\ \hline 4x^2 & - & 3x & - & 2 \\ \hline \end{array}

∴ The required expression is 4x2 - 3x - 2

(iii) What should be added to m2 - 2mn + 5n2 to get n2 + mn - m2?

To find what must be added to A to get B, we simply calculate B - A.

∴ Subtract m2 - 2mn + 5n2 from n2 + mn - m2

Arranging the terms to match (m2, mn, n2):

m2+mn+n2+m22mn+5n2m2+2m2+3mn4n2\begin{array}{rcccc} -m^2 & + & mn & + & n^2 \\ +m^2 & - & 2mn & + & 5n^2 \\ -\phantom{m^2} & + & & - \\ \hline -2m^2 & + & 3mn & - & 4n^2 \\ \hline \end{array}

∴ The required expression is -2m2 + 3mn - 4n2

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