If x raise to power two +14x +k is a perfect square find k
x^2 + 14x + k
= x^2 + 14x + 49 + k - 49
= (x+7)^2 + k-49
if (x+7)^2 = x^2 + 14x + k , k-49 = 0 or k = 49
or
(x + a)^2 = x^2 + 2ax + a^2
compare with x^2 + 14x + k, then
2ax = 14x
2a = 14
a = 7 and k = a^2 = 49
or, by just knowing the result of (a + b)^2
x^2 + 14x + k, take half of 14, then square it.
Knock, knock.
Who's there?
Quadratic.
Quadratic who?
Quadratic formula to the rescue!
To find the value of k, we can use the fact that the given equation is a perfect square trinomial. The perfect square trinomial can be written as (x + m)^2, where m is half of the coefficient of x.
In this case, since the coefficient of x is 14, half of it would be 7.
So, (x + 7)^2 equals x^2 + 14x + 49.
Comparing this with the given equation, we can see that k must be 49.
Thus, the value of k is 49.
To determine the value of k such that the expression x^2 + 14x + k is a perfect square, we need to consider the properties of perfect square trinomials.
A perfect square trinomial can be written in the form (a + b)^2, where a and b are constants. In this case, we have x^2 + 14x + k as the given expression.
To find the middle term coefficient (14x), we take half of it (which is 7) and square it (49). Therefore, the value of k that makes the expression a perfect square is 49.
So, k = 49.
To find the value of k such that the expression x^2 + 14x + k is a perfect square, we need to determine the perfect square trinomial that it represents.
The general form of a perfect square trinomial is (a + b)^2 = a^2 + 2ab + b^2.
Comparing this to the given expression, we can see that a^2 = x^2 and 2ab = 14x.
From this information, we can determine that a = x and 2ab = 14x. Simplifying, we find:
2ab = 14x
2x * x = 14x
2x^2 = 14x
Now we can solve for x by dividing both sides of the equation by 2:
x^2 = 7x
Next, subtracting 7x from both sides:
x^2 - 7x = 0
Now we have a quadratic equation which we can factor:
x(x - 7) = 0
From this equation, we have two possible solutions for x: x = 0 or x = 7.
If we substitute x = 0 into the given expression x^2 + 14x + k, we get:
0^2 + 14(0) + k
0 + 0 + k
k
Therefore, when x = 0, k = 0.
If we substitute x = 7 into the given expression x^2 + 14x + k, we get:
7^2 + 14(7) + k
49 + 98 + k
147 + k
So, when x = 7, k = 147.
Therefore, the value of k that makes the expression x^2 + 14x + k a perfect square is k = 147.