given that 2xsquared-5x+k is a perfect square . find the value of k
2x^2 - 5x + k
What we can do here is to get the discriminant. For a quadratic equation with form ax^2 + bx + c = 0, the discriminant is equal to
b^2 - 4ac
The value of discriminant for a perfect square (which therefore has equal or double roots) is equal to zero. Substituting,
b^2 - 4ac = 0
(-5)^2 - 4(2)(k) = 0
25 - 8k = 0
-8k = -25
k = 25/8
Hope this helps~ :)
To determine the value of k in the expression 2x^2 - 5x + k, we need to check if it can be a perfect square trinomial.
A perfect square trinomial has the form (a + b)^2. Using this pattern, we can rewrite 2x^2 - 5x + k as (sqrt(2)x - sqrt(k))^2.
Expanding (sqrt(2)x - sqrt(k))^2, we get:
(sqrt(2)x)^2 - 2(sqrt(2)x)(sqrt(k)) + (sqrt(k))^2
2x^2 - 2(sqrt(2xk)) + k
Comparing the expanded form with our original expression, we find that the coefficient of x^2 matches, the coefficient of x matches, but the constant term doesn't match.
The constant term is k in the original expression and k^2 in the expanded form. Therefore, to make the expression a perfect square, we need to set k^2 equal to the constant term, which is k.
So, we have the equation: k^2 = k
To solve this equation, we can subtract k from both sides:
k^2 - k = 0
Factoring out k gives us:
k(k - 1) = 0
Setting each factor equal to zero:
k = 0
k - 1 = 0
Therefore, the possible values for k are:
k = 0
k = 1
To find the value of k, we need to rewrite the given quadratic expression as a perfect square trinomial.
The general form of a perfect square trinomial is (a + b)^2 = a^2 + 2ab + b^2. In this case, our quadratic expression is 2x^2 - 5x + k.
Since the coefficient of the x^2 term is 2, we can take the square root of the coefficient to find a. So, a = sqrt(2).
Now, let's expand (a + b)^2 = a^2 + 2ab + b^2 by substituting the values of a and b:
(√2)^2 + 2(√2)b + b^2 = 2x^2 - 5x + k
2 + 2(√2)b + b^2 = 2x^2 - 5x + k
Simplifying this equation, we get:
2 + 2√2b + b^2 = 2x^2 - 5x + k
Now, we need to match the coefficients of x^2, x, and the constant term on both sides of the equation.
From the given quadratic expression, we see that the coefficient of x^2 is 2, which means a^2 = 2. Therefore, a = √2.
By comparing coefficients, we get:
-5 = 2√2b
k = 2 + b^2
To solve for b, we divide both sides of the first equation by 2√2:
-5/(2√2) = b
Simplifying this, we get:
b = -5/(2√2) = -5√2/4 = -5√2/2√2 * 1/2 = -5/2√2 * √2/√2 = -5√2/4
Now, substitute the value of b into the equation for k:
k = 2 + b^2
k = 2 + (-5√2/4)^2
k = 2 + 25*2/4
k = 2 + 50/4
k = 2 + 25/2
k = 2 + 12.5
k = 14.5
Therefore, the value of k is 14.5.