Consider the equation x(square) + 4px + 2q = 0 where p and q are real constants.
a) by completing square, show that (x+2p)square =4P(SQUARE)-2q
b) Hence show that x= -2p +/- square root(4 p(square) -2p )
c) Use the above results to solve the equations
1) x(square) + 8x +14 =0
2) x(square) -16x -6 =0
x^2 + 4px + 2q = 0
x^2 + 4px + (2p)^2 + 2q - (2p)^2 = 0
(x+2p)^2 = 4p^2 - 2q
b follows immediately
c and d just use part b to evaluate the roots.
Evaluate square root of 25+x square dx
a) To complete the square, we follow these steps:
1. Start with the equation x^2 + 4px + 2q = 0.
2. Move the constant term, 2q, to the right side of the equation: x^2 + 4px = -2q.
3. Add the square of half the coefficient of x (which is p) to both sides of the equation:
x^2 + 4px + p^2 = -2q + p^2.
4. Rearrange the terms on the right side:
x^2 + 4px + p^2 = p^2 - 2q.
5. Factor the left side of the equation as a perfect square:
(x + 2p)^2 = p^2 - 2q.
Therefore, (x + 2p)^2 = 4p^2 - 2q.
b) Using the result from part (a), we have:
(x + 2p)^2 = 4p^2 - 2q.
Taking the square root of both sides, we get:
x + 2p = ± √(4p^2 - 2q).
Subtracting 2p from both sides gives us:
x = -2p ± √(4p^2 - 2q).
c) Let's solve the equations using the derived formula:
1) For x^2 + 8x + 14 = 0, we identify p = 4 and q = 7. Plugging the values into the formula:
x = -2(4) ± √(4(4^2) - 2(7)).
= -8 ± √(64 - 14).
= -8 ± √50.
Therefore, x = -8 ± √50.
2) For x^2 - 16x - 6 = 0, we identify p = -8 and q = -3. Plugging the values into the formula:
x = -2(-8) ± √(4((-8)^2) - 2(-3)).
= 16 ± √(256 + 6).
= 16 ± √262.
Therefore, x = 16 ± √262.
a) To complete the square, we start with the equation x^2 + 4px + 2q = 0.
Step 1: Move the constant term (2q) to the right side:
x^2 + 4px = -2q
Step 2: Group the x terms and complete the square:
x^2 + 4px + (4p/2)^2 = -2q + (4p/2)^2
x^2 + 4px + (2p)^2 = -2q + 4p^2/4
x^2 + 4px + 4p^2 = -2q + 4p^2/4
(x + 2p)^2 = -2q + 4p^2/4
(x + 2p)^2 = -2q + p^2
b) Now that we have (x + 2p)^2 = -2q + p^2, we can solve for x.
Taking the square root of both sides, we get:
x + 2p = ±√(-2q + p^2)
Simplifying further:
x = -2p ± √(-2q + p^2)
c) Now let's solve the given equations using the above results.
1) For the equation x^2 + 8x + 14 = 0:
Comparing this equation with the standard form (x + 2p)^2 = 4p^2 - 2q,
We have:
2p = 8
p = 4
-2q = 14
q = -7
Substituting the values of p and q:
x = -2(4) ± √(4(4)^2 - 2(-7))
x = -8 ± √(64 + 14)
x = -8 ± √78
So the solutions are:
x = -8 + √78
x = -8 - √78
2) For the equation x^2 - 16x - 6 = 0:
Comparing this equation with the standard form (x + 2p)^2 = 4p^2 - 2q,
We have:
2p = -16
p = -8
-2q = -6
q = 3
Substituting the values of p and q:
x = -2(-8) ± √(4(-8)^2 - 2(3))
x = 16 ± √(256 - 6)
x = 16 ± √250
x = 16 ± 5√10
So the solutions are:
x = 16 + 5√10
x = 16 - 5√10
These are the values of x that satisfy the given equations.