Using the method completing the square. Find the roots of this equation 4d^2= (d+4)(-d+12)
4d^2= (d+4)(-d+12)
4d^2 = -d^2 + 12d - 4d + 48
5d^2 - 8dd = 48
d^2 - (8/5)d = 48/5
d^2 - (8/5)d + 16/25 = 48/5 + 16/25
(d - 4/5)^2 = 256/25
d - 4/5 = ± 16/5
d = 4/5 + 16/5 OR d = 4/5 - 16/5
d = 4 or d = -12/5
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To find the roots of the equation 4d^2 = (d + 4)(-d + 12) using the method of completing the square, follow these steps:
Step 1: Expand the equation.
4d^2 = -d^2 + 8d + 48
Step 2: Move all terms to one side to have a quadratic equation equal to zero.
4d^2 + d^2 - 8d - 48 = 0
Step 3: Combine like terms.
5d^2 - 8d - 48 = 0
Step 4: Divide the entire equation by the leading coefficient (in this case, 5) to simplify the equation.
d^2 - 8/5d - 48/5 = 0
Step 5: Complete the square by adding the square of half the coefficient of the linear term (in this case, -8/5) to both sides of the equation.
d^2 - 8/5d + (-8/5)^2 - 48/5 + (-8/5)^2 = (8/5)^2
d^2 - 8/5d + 64/25 - 48/5 + 64/25 = 64/25
Step 6: Simplify the equation.
d^2 - 8/5d + 16/25 - 240/25 + 64/25 = 64/25
d^2 - 8/5d - 160/25 + 64/25 = 64/25
d^2 - 8/5d - 96/25 = 64/25
Step 7: Combine like terms.
d^2 - 8/5d - 96/25 = 0
Step 8: Factorize the quadratic equation, if possible.
(d - 16/5)(d + 6/5) = 0
Step 9: Set each factor equal to zero and solve for d.
d - 16/5 = 0 --> d = 16/5
d + 6/5 = 0 --> d = -6/5
Therefore, the roots of the equation 4d^2 = (d + 4)(-d + 12) are d = 16/5 and d = -6/5.