1. ## equations and quadratic functions

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.

2. 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 $\displaystyle \geq -2$.

3. Originally Posted by lra11
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