Given that $\frac{5a-3b}{2a} = \frac{4}{3}$, find the value of $\frac{a}{b}$. Answer ... Express $x^2 - 6x + 1$ in the form $(x + a)^2 + b$. Answer ... Hence solve the equation $x^2 - 6x + 1 = 0$, giving your answers correct to two decimal places. Answer x = ... or x = ... Given that 4500 workers, each working 8 hours a day, will complete the Thomson Line in 1800 days.
If 4800 workers work on the project with each worker working for 10 hours a day, find the number of days it would take to complete the project. Answer ... days

1. Clear the fractions by cross-multiplication.

$\frac{5a-3b}{2a}=\frac{4}{3}$
$3(5a-3b)=4(2a)$

2. Rearrange to find the ratio $\frac{a}{b}$.

$15a-9b=8a$
$7a=9b$
$\frac{a}{b}=\frac{9}{7}$

1. Complete the square for the quadratic expression.

$x^2-6x+1=(x-3)^2-9+1=(x-3)^2-8$

1. Substitute the completed-square form into the equation.

From part (b)(i),
$x^2-6x+1=(x-3)^2-8.$
So
$(x-3)^2-8=0.$

2. Solve the equation by isolating the squared term and taking square roots.

$(x-3)^2=8$
$x-3=\pm\sqrt{8}$
$x=3\pm\sqrt{8}.$

3. Evaluate both roots and round to two decimal places.

$x=3+\sqrt{8}=5.828\ldots$
$x=3-\sqrt{8}=0.171\ldots$
Therefore, to two decimal places,
$x=5.83\text{ or }x=0.17.$

1. Find the total amount of work in worker-hours.

Total work $=4500\times 8\times 1800$ worker-hours.

2. Divide the same total work by the new daily work rate to find the number of days.

Let the required number of days be $d$.

$4800\times 10\times d=4500\times 8\times 1800$

$d=\frac{4500\times 8\times 1800}{4800\times 10}=1350$