MarkFL said:Begin by stating the roots of the quadratic, using the quadratic formula...what do you have?
RTCNTC said:Discriminant?
MarkFL said:No, the roots themselves. Recall, if:
\(\displaystyle ax^2+bx+c=0\)
then:
\(\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
What are the roots of the given quadratic?
RTCNTC said:The roots are not given. The roots differ by 2. This is the only data given.
MarkFL said:True, the roots are no given, but you can compute them...we should begin by equating the given quadratic to zero:
\(\displaystyle 2x^2+5x–k=0\)
Using the quadratic formula, we find the roots to be:
\(\displaystyle x=\frac{-5\pm\sqrt{5^2-4(2)(-k)}}{2(2)}=\frac{-5\pm\sqrt{25+8k}}{4}\)
Now, if the roots are to differ by 2, then take the larger root, subtract the smaller, and set that difference equal to 2:
\(\displaystyle \frac{-5+\sqrt{25+8k}}{4}-\frac{-5-\sqrt{25+8k}}{4}=2\)
Now solve this for $k$...
Another approach would be to write:
\(\displaystyle 2x^2+5x–k=2(x-(r+2))(x-r)=2x^2-4(r+1)x+2r(r+ 2)\)
Equating coefficients, we obtain:
\(\displaystyle -4(r+1)=5\)
\(\displaystyle k=-2r(r+ 2)\)
Solve the first to get $r$, and then use that value of $r$ in the second to determine $k$. :D
topsquark said:As MarkFL commented, you need to solve the following for k:
\(\displaystyle \frac{-5+\sqrt{25+8k}}{4}-\frac{-5-\sqrt{25+8k}}{4}=2\)
\(\displaystyle (-5+\sqrt{25+8k}) - (-5-\sqrt{25+8k}) = 4 \cdot 2 = 8\)
\(\displaystyle (-5 - (-5)) + (\sqrt{25 + 8k} - - \sqrt{25 + 8k}) = 8\)
\(\displaystyle 2~\sqrt{25 + 8k} = 8\)
Can you finish from here?
-Dan
MarkFL said:True, the roots are no given, but you can compute them...we should begin by equating the given quadratic to zero:
\(\displaystyle 2x^2+5x–k=0\)
Using the quadratic formula, we find the roots to be:
\(\displaystyle x=\frac{-5\pm\sqrt{5^2-4(2)(-k)}}{2(2)}=\frac{-5\pm\sqrt{25+8k}}{4}\)
Now, if the roots are to differ by 2, then take the larger root, subtract the smaller, and set that difference equal to 2:
\(\displaystyle \frac{-5+\sqrt{25+8k}}{4}-\frac{-5-\sqrt{25+8k}}{4}=2\)
Now solve this for $k$...
Another approach would be to write:
\(\displaystyle 2x^2+5x–k=2(x-(r+2))(x-r)=2x^2-4(r+1)x+2r(r+ 2)\)
Equating coefficients, we obtain:
\(\displaystyle -4(r+1)=5\)
\(\displaystyle k=-2r(r+ 2)\)
Solve the first to get $r$, and then use that value of $r$ in the second to determine $k$. :D
Finding the value of k is a common task in scientific research, especially in fields such as physics and mathematics. It involves determining the numerical value of a constant, known as k, that is used in a mathematical equation or formula.
Knowing the value of k allows scientists to accurately predict and model various phenomena, as well as make informed decisions based on the results of their research. It also helps to establish the relationship between different variables in an equation.
There are various methods for determining the value of k, depending on the specific equation or formula being used. Some methods include experimentation, data analysis, and mathematical calculations.
Yes, the value of k can change depending on the specific conditions or variables involved in a particular equation or formula. It can also vary between different contexts and applications.
One common mistake when finding the value of k is using incorrect or incomplete data. It is essential to ensure that all relevant variables are considered and that the data used is accurate and reliable. Another mistake is using the incorrect method for determining the value of k, which can lead to inaccurate results.