Solving the quadratic equation $3t^2-788t-7335=0$ by hand

Question

I was solving $\sqrt{3t+52}+\sqrt{2t+162}=\sqrt{-t+280}$, and after some squaring and rearranging I got the following quadratic equation: $$3t^2-788t+(9\times121-52\times162)=0$$ which is equivalent to: $$3t^2-788t-7335=0$$ My goal is to solve the above equation by hand, without any calculator or software. I tried multiplying the equation by $3$ and extracting $3t$:

$$(3t)^2-788\times(3t)-7335\times3=0$$ And factoring this into $(3t-a)(3t-b)=0$ gives \begin{cases} {a+b=788} \\ {ab=-7335\times3} \end{cases} I'm not sure if this is helpful, but I'm also interested in other methods for solving the equation by hand.

Edit:

I also tried quadratic formula which gives$\dfrac{788\pm\sqrt{708964}}6$, but I couldn't find the value of $\sqrt{708964}$ .

The quadratic formula is "by hand" and does the job. It is not clear what more you want — Anne Bauval, Feb 08 '25 at 07:12
This question is similar to: Why can ALL quadratic equations be solved by the quadratic formula?. If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. — Ak3.14, Feb 08 '25 at 07:15
A simple search on how to compute square roots by hand does the job. — Ak3.14, Feb 08 '25 at 07:21
OP, 708964 is actually $ 2^2 \times 421^2 $, so $ \sqrt{708964}=\sqrt{2^2\times 421^2}=2\times 421=842$. Try prime factorising the number inside the square root. — ThankYouForFlyingRyanair, Feb 08 '25 at 07:38

aarbee · Accepted Answer · 2025-02-08T07:25:25.477

3

$7335\times3=5\times1467\times3=5\times9\times163\times3=815\times27$

And $815-27=788$

If you want to use quadratic formula then you can see that $(800)^2=640000, (840)^2=(800+40)^2=705600, (842)^2=708964$

edited Feb 08 '25 at 07:25

answered Feb 08 '25 at 07:18

aarbee

10,749

ThankYouForFlyingRyanair · Answer 2 · 2025-02-08T07:35:31.213

You can use at alternative to the quadratic formula, that is by using the factor theorem, which states that, if $f(t)=3t^2-788t-7335$ and $f(c)=0$, then $x-c$ is a factor of $f(t)$. Divide $\frac{f(t)}{x-c}$ to find the other factor of $f(t)$.

When you use $c=9$, you will see that $f(-9)=0$, which shows that $x+9$ is a factor of $f(t)$. After that, divide $f(t)$ by $x+9$ to get the other factor which is $3x-815$.

So the two factors of $f(t)$ are the two solutions to $f(t)$, which are $x=-9$ and $x=\frac{815}{3}$.

Solving the quadratic equation $3t^2-788t-7335=0$ by hand

2 Answers2