Solve the equation by completing the square. around to the nearest hundredth if necessary.
X^2+5x=6
To complete the square, we need to add and subtract (5/2)^2 = 6.25 inside the parentheses:
x^2 + 5x + 6.25 - 6.25 = 6
Now we can write the left side as a perfect square:
(x + 2.5)^2 - 6.25 = 6
Add 6.25 to both sides:
(x + 2.5)^2 = 12.25
Take the square root of both sides (remembering to include the ±):
x + 2.5 = ±3.5
Subtract 2.5 from both sides:
x = -2.5 ± 3.5
Which gives us two solutions: x = 1 or x = -6.
To solve the equation by completing the square, follow these steps:
Step 1: Move the constant term to the right side of the equation:
X^2 + 5x = 6 becomes X^2 + 5x - 6 = 0.
Step 2: Divide the coefficient of x by 2, square it, and add it to both sides of the equation:
X^2 + 5x + (5/2)^2 - (5/2)^2 - 6 = (5/2)^2.
This simplifies to: X^2 + 5x + 25/4 - 25/4 - 6 = 25/4.
Step 3: Simplify the equation:
X^2 + 5x + 25/4 - 25/4 - 6 = 25/4 becomes
(X + 5/2)^2 - 31/4 = 25/4.
Step 4: Move the constant term to the right side of the equation and simplify:
(X + 5/2)^2 = 25/4 + 31/4,
(X + 5/2)^2 = 56/4,
(X + 5/2)^2 = 14.
Step 5: Take the square root of both sides:
X + 5/2 = ±√14.
Step 6: Solve for x:
X = -5/2 ± √14.
Therefore, the solutions to the equation are:
X = -5/2 + √14 ≈ 0.87 (rounded to the nearest hundredth),
X = -5/2 - √14 ≈ -5.87 (rounded to the nearest hundredth).