What Must Be Added To x^2 + 10x + 23 To Make It A Perfect Square
write
x^2 + 10 x
add half of ten squared
5^2 = 25
so
x^2 + 10 x + 25 is a perfect square
so we need to add 2
to 23
to make 25
and then
(x+5)^2 = x^2 + 10 x + 25
To make the quadratic expression x^2 + 10x + 23 a perfect square, we need to add a specific term.
Let's start by rewriting the given expression in the form of a perfect square.
We know that a perfect square trinomial is of the form (x + a)^2, where 'a' is a constant. Expanding this square, we get:
(x + a)(x + a) = x^2 + 2ax + a^2
Comparing this with the given expression, x^2 + 10x + 23, we can see that the quadratic term (2ax) is currently missing in the given expression. So, we need to find a value for 'a' that will give us 10x when multiplied by 2a.
In this case, 2a = 10
Solving for 'a', we divide both sides of the equation by 2:
a = 5
Now, we know that the perfect square form of x^2 + 10x + 23 is (x + 5)^2.
To confirm, we can expand (x + 5)^2:
(x + 5)(x + 5) = x^2 + 5x + 5x + 25 = x^2 + 10x + 25
As you can see, when we expand (x + 5)^2, we get the original quadratic expression, x^2 + 10x + 23, plus 2ax, which in this case is 2 * 5x = 10x.
Therefore, the term that must be added to x^2 + 10x + 23 to make it a perfect square is 25.