Tadayen said:
A tablet computer has a 7" diagonal screen. The length of the screen is 2.7 " longer than the width. Find the dimensions of the screen.
I will use height rather than length, and this makes more sense to me. Let's draw a diagram first (where all measures are in inches):
View attachment 5036
We are given that the height $h$ is 2.7" more than the width $w$, so we may state:
$$h=w+2.7\tag{1}$$
And by the Pythagorean theorem, we may write:
$$w^2+h^2=7^2\tag{2}$$
Now, using (1) we may substitute for $h$ in (2) to get:
$$w^2+(w+2.7)^2=7^2$$
I don't like working with decimals, so let's instead write:
$$w^2+\left(w+\frac{27}{10}\right)^2=7^2$$
Adding within the parentheses, we have:
$$w^2+\left(\frac{10w+27}{10}\right)^2=7^2$$
Multiplying through by $10^2$, we obtain:
$$(10w)^2+(10w+27)^2=(7\cdot10)^2$$
Squaring the binomial on the left, we get:
$$(10w)^2+(10w)^2+2(10w)(27)+27^2=(70)^2$$
Simplify further:
$$2(10w)^2+54(10w)+\left(27^2-70^2\right)=0$$
Factor difference of squares:
$$2(10w)^2+54(10w)+(27+70)(27-70)=0$$
$$2(10w)^2+54(10w)-97\cdot43=0$$
$$2(10w)^2+54(10w)-4171=0$$
Let $u=10w$, and we have a quadratic in $u$ in standard form:
$$2u^2+54u-4171=0$$
Applying the quadratic formula (and discarding the negative root), we obtain:
$$u=\frac{-54+\sqrt{54^2+4(2)(4171)}}{2(2)}=\frac{-2(27)+2\sqrt{27^2+(2)(4171)}}{2(2)}=\frac{-27+\sqrt{729+8342}}{2}=\frac{-27+\sqrt{9071}}{2}$$
Hence:
$$10w=\frac{-27+\sqrt{9071}}{2}\implies w=\frac{-27+\sqrt{9071}}{20}$$
And so:
$$h=\frac{-27+\sqrt{9071}}{20}+\frac{27}{10}=\frac{-27+\sqrt{9071}+54}{20}=\frac{27+\sqrt{9071}}{20}$$