+1245 There are two distinct solutions $x$ to the equation $18+5x^2=20x$. If each solution is rounded to the nearest integer, and then these two integers are multiplied together, what is the result?
+2446 In order to answer this question, one must determine what the solutions are to the equation \(18+5x^2=20x\). Let's do that, shall we?
| \(18+5x^2=20x\) | When solving quadratic equations, it is generally advisable to move all terms to one side of the equation. |
| \(5x^2-20x+18=0\) | This equation is probably best solved with the quadratic formula. |
| \(a=5; b=-20; c=18\\ x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) | Now, plug into this equation and find the roots. |
| \(x_{1,2}=\frac{-(-20)\pm\sqrt{(-20)^2-4*5*18}}{2*5}\) | The rest is a matter of simplifying accurately. |
| \(x_{1,2}=\frac{20\pm\sqrt{40}}{10}\) | Generally, there is a need to put radicals into simplest form, but this is not necessary since the question asks for their rounded value. |
| \(x_1\approx3\\ x_2\approx1\) | Let's find the rounded roots' product. |
| \(x_1*x_2=3*1=3\) | This is it! |
