Solve the equation 5x^2-5x-63=0 by using the "complete the square" method. Give your answer to 2 decimal place.
Please show me in clear steps how to get the answers 4.08 and -3.08.
5x^2-5x-63
= 5(x^2-x) - 63
Recall that (x-a)^2 = a^2-2a+a^2
Here we have 2a = 1, so a = 1/2. So, add and subtract 1/4 * 5
= 5(x^2-x + 1/4) - 63 - 5/4
= 5(x - 1/2)^2 - 267/4
To get the roots, set that to zero:
5(x - 1/2)^2 - 257/4 = 0
5(x - 1/2)^2 = 257/4
(x - 1/2)^2 = 257/20
x - 1/2 = ±√(257/20)
x = 1/2 ±√(257/20)
x = -3.08 or 4.08
xy=90
2x+y=36
To solve the equation 5x^2 - 5x - 63 = 0 by completing the square, follow these steps:
Step 1: Divide the entire equation by the leading coefficient (5). This step is necessary to simplify the equation before completing the square. Divide all terms by 5 to get:
x^2 - x -63/5 = 0
Step 2: Move the constant term (63/5) to the other side of the equation by adding it to both sides. This will allow us to complete the square on the quadratic expression. The equation becomes:
x^2 - x = 63/5
Step 3: Add the square of half of the coefficient of x to both sides. The coefficient of x is -1. Half of -1 is -1/2, and the square of -1/2 is 1/4. Add 1/4 to both sides to complete the square on the left side:
x^2 - x + 1/4 = 63/5 + 1/4
Step 4: Simplify both sides of the equation:
(x - 1/2)^2 = 63/5 + 1/4
Step 5: Find a common denominator and add the fractions on the right side:
(x - 1/2)^2 = (252 + 5)/20
(x - 1/2)^2 = 257/20
Step 6: Take the square root of both sides to solve for x. Remember to consider both the positive and negative square root:
x - 1/2 = ±√(257/20)
Step 7: Isolate x by adding 1/2 to both sides:
x = 1/2 ±√(257/20)
Step 8: Simplify the radical expression under the square root:
x ≈ 1/2 ± 4.03
Step 9: Round the two decimal places:
x ≈ 1/2 + 4.03 ≈ 4.08
x ≈ 1/2 - 4.03 ≈ -3.08
Therefore, the solutions to the equation 5x^2 - 5x - 63 = 0, using the complete the square method, are approximately 4.08 and -3.08.