The equation x ^ 2 - 8x - 5 = 0 can be transformed into the equation (x - p) ^ 2 = q where p and q are real numbers. What is the value of q?
We know that for any quadratic equation of the form x^2 + bx + c, the expression (x - b/2)^2 - (b^2/4) is equal to it. So, to transform x^2 - 8x - 5 = 0 into the form (x - p)^2 = q, we need to complete the square by adding a term that will allow us to factor a perfect square trinomial.
First, we need to factor out any common factors of the quadratic equation, so:
x^2 - 8x - 5 = 0
(x - 4)^2 - 21 = 0
Now we have the expression in the form (x - p)^2 - q, where p = 4 and q = -21. However, we want q to be a positive number, so we can multiply the entire equation by -1 to get:
-(x - 4)^2 + 21 = 0
(x - 4)^2 = 21
Therefore, the value of q is 21.
To transform the equation x^2 - 8x - 5 = 0 into the equation (x - p)^2 = q, we will use the method of completing the square.
Step 1: Move the constant term (-5) to the right-hand side of the equation:
x^2 - 8x = 5
Step 2: To complete the square, take half of the coefficient of x and square it. In this case, half of -8 is -4, and (-4)^2 is 16. Add 16 to both sides of the equation:
x^2 - 8x + 16 = 5 + 16
Step 3: Simplify both sides:
(x - 4)^2 = 21
Now, we have transformed the equation into the form (x - p)^2 = q. The value of q is 21.