Question

Factorize the following expression: $10a^2-15b^2+20c^2$

Answer 1

Step 1: Find factors of $\mathbf{10}{\mathit{a}}^{\mathbf{2}}$

$10a^2=2\times 5\times a\times a$ …..[1]

Step 2: Find factors of $\mathbf{15}{\mathit{b}}^{\mathbf{2}}$

$15b^2=3\times 5\times b\times b$ ….[2]

Step 3: Find factors of $\mathbf{20}{\mathit{c}}^{\mathbf{2}}$

$20c^2=2\times 2\times 5\times c\times c$ ….[3]

Step 4: Find common factors of $\mathbf{10}{\mathit{a}}^{\mathbf{2}}\mathbf{-}\mathbf{15}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{20}{\mathit{c}}^{\mathbf{2}}$

$10a^2-15b^2+20c^2$ $=$$\left(2\times 5\times a\times a\right)$$-\left(3\times 5\times b\times b\right)$$+\left(2\times 2\times 5\times c\times c\right)$

$=5(2{\mathrm{a}}^2-3{\mathrm{b}}^2+4{\mathrm{c}}^2)$

Hence, $\mathbf{10}{\mathit{a}}^{\mathbf{2}}\mathbf{-}\mathbf{15}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{20}{\mathit{c}}^{\mathbf{2}}$$\mathbf{=}\mathbf{5}\mathbf{(}\mathbf{2}{\mathbf{a}}^{\mathbf{2}}\mathbf{-}\mathbf{3}{\mathbf{b}}^{\mathbf{2}}\mathbf{+}\mathbf{4}{\mathbf{c}}^{\mathbf{2}}\mathbf{)}$