Please help, I'm really confused.
1) If x² + mx + n is a perfect square, show that m²=4n
and
2) find value of k
x² + (2k+10)x + k² + 5 = 0
Help would be much appreciated. Thanks.
Hello, lra11!
1) If $\displaystyle x^2 + mx + n$ is a perfect square, show that: .$\displaystyle m^2 = 4n$
If $\displaystyle x^2+mx+n$ is a square, the equation $\displaystyle x^2 + mx + n \:=\:0$ has one root.
If a quadratic equation has one root, its discriminant $\displaystyle (b^2-4ac)$ is zero.
Hence: .$\displaystyle m^2 - 4n \:=\:0$
Therefore: .$\displaystyle m^2\:=\:4n$
2) Find value of $\displaystyle k$
. . $\displaystyle x^2 + (2k+10)x + k^2 + 5 \:= \:0$
What are the conditions for this quadratic equation?
As given, $\displaystyle k$ can be any number $\displaystyle \geq -2$.
remember what you do when you are completing the square? if a quadratic is a complete square, it means that the constant is the square of 1/2 of the coefficient of x. thus, if $\displaystyle x^2 + mx + n$ is a complete square, it follows immediately that $\displaystyle n = \left( \frac m2 \right)^2$ and the rest is trivial