Question

a) A positive number is divided into two parts, such that the sum of the squares of the two parts is 208. The square of the larger part is 8 times the smaller part. Taking $x$ as the smaller part of the two parts, find the number.

b) Solve the following quadratic equation and give the answer correct to two significant figures:
$4x^2-7x+2=0$
[6 MARKS]

Answer 1

Each subpart: 3 Marks

a) Let smaller part be $x$.
Then, $(largerpart{)}^2=8x$
According to the question,
$8x+x^2=208\Rightarrow x^2+8x-208=0$
Now, $x=\frac{-8\pm \sqrt{8^2-4\times 1\times (-208)}}{2\times 1}=\frac{-8\pm \sqrt{64+832}}{2}=\frac{-8\pm 29.93}{2}Eitherx=\frac{-8+29.93}{2}orx=\frac{-8-29.93}{2}\Rightarrow x=\frac{21.93}{2}=10.97orx=\frac{-37.93}{2}=-18.97$
Here, if the smaller part is 10.97, the larger part = 9.36, which is not possible. If smaller part is -18.97, the larger part = $\sqrt{-151.76}$ which is not possible.
So, such numbers do not exist.

b)
We have, $4x^2-7x+2=0$
Using the quadratic formula
we get: x = $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
Now, $x=\frac{7\pm \sqrt{49-4\times 4\times 2}}{2\times 4}=\frac{7\pm \sqrt{49-32}}{8}=\frac{7\pm \sqrt{17}}{8}=\frac{7\pm 4.123}{8}Eitherx=\frac{7+4.123}{8}orx=\frac{7-4.123}{8}\Rightarrow x=\frac{11.123}{8}orx=\frac{2.877}{8}\Rightarrow x=1.390orx=0.359=0.36$
Hence, x = 1.4 or 0.36 (upto two significance figures).