Question

Fill in the blanks.
(i) x² − 18x + 81 = (......)
(ii) 4 − 36x² = (......)(......)(......)
(iii) x² − 14x + 13 = (......)(......)
(iv) 9z² − x² − 4y² + 4xy = (......)(......)
(v) abc − ab − c + 1 = (......)(......)

Answer 1

(i) ${\left(x-9\right)}^2$

$$\begin{gathered} x^2-18x+81 \\ =x^2-2\times x\times 9+9^2 \\ ={\left(x-9\right)}^2 \end{gathered}$$


(ii) $\left(4\right)\left(1-3x\right)\left(1+3x\right)$

$$\begin{gathered} 4-36x^2 \\ =4\left(1-9x^2\right) \\ =4\left(1^2-{\left(3x\right)}^2\right) \\ =\left(4\right)\left(1-3x\right)\left(1+3x\right) \end{gathered}$$


(iii) $\left(x-13\right)\left(x-1\right)$

$$\begin{gathered} x^2-14x+13 \\ =x^2-13x-x+13 \\ =x\left(x-13\right)-1\left(x-13\right) \\ =\left(x-13\right)\left(x-1\right) \end{gathered}$$


(iv) $\left(3z+x-2y\right)\left(3z-x+2y\right)$

$$\begin{gathered} 9z^2-x^2-4y^2+4xy \\ =9z^2-\left(x^2-4xy+4y^2\right) \\ =9z^2-\left(x^2-2\times x\times 2y+{\left(2y\right)}^2\right) \\ =9z^2-{\left(x-2y\right)}^2 \\ ={\left(3z\right)}^2-{\left(x-2y\right)}^2 \\ =\left(3z+\left(x-2y\right)\right)\left(3z-\left(x-2y\right)\right) \\ =\left(3z+x-2y\right)\left(3z-x+2y\right) \end{gathered}$$


(v) $\left(c-1\right)\left(ab-1\right)$

$$\begin{gathered} abc-ab-c+1 \\ =ab\left(c-1\right)-1\left(c-1\right) \\ =\left(c-1\right)\left(ab-1\right) \end{gathered}$$