Solve by completing the square: x^2+3x-28=0
x^2+3x-28 = 0
x^2+3x = 28
x^2 + 2(3/2)x + (3/2)^2 = 28 + (3/2)^2
(x + 3/2)^2 = 121/4 = (11/2)^2
x + 3/2 = ± 11/2
x = -3/2 ± 11/2
x = -7,4
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Why did the square go to the party alone? Because it couldn't find its plus one! Let's complete the square for the equation x^2+3x-28=0:
Step 1: Move the constant term to the other side: x^2+3x=28
Step 2: Take half of the coefficient of x and square it: (3/2)^2 = 9/4
Step 3: Add this value to both sides of the equation:
x^2+3x+9/4=28+9/4
Step 4: Factor the left side as a perfect square:
(x+3/2)^2 = 112/4 + 9/4
(x+3/2)^2 = 121/4
Step 5: Take the square root of both sides:
x + 3/2 = ±√(121/4)
x + 3/2 = ±11/2
Step 6: Solve for x:
x = -3/2 ± 11/2
Thus, the solutions to the equation x^2+3x-28=0 are x = -7 and x = 4.
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
x^2 + 3x = 28
Step 2: Take half of the coefficient of x (which is 3), square it, and add it to both sides of the equation:
x^2 + 3x + (3/2)^2 = 28 + (3/2)^2
(x + 3/2)^2 = 28 + 9/4
Step 3: Simplify the right side of the equation:
(x + 3/2)^2 = 112/4 + 9/4
(x + 3/2)^2 = 121/4
Step 4: Take the square root of both sides, considering both positive and negative roots:
x + 3/2 = ±√(121/4)
x + 3/2 = ±11/2
Step 5: Solve for x:
For x + 3/2 = 11/2:
x = 11/2 - 3/2
x = 8/2
x = 4
For x + 3/2 = -11/2:
x = -11/2 - 3/2
x = -14/2
x = -7
Therefore, the solutions to the equation x^2 + 3x - 28 = 0 are x = 4 and x = -7.