2x^2+x-28=0
Solve by completing the square
2x^2+x = 28
divide by 2
x^2 + x/2 = 14
add 1/16 to both sides
x^2 + x/2 + 1/16 = 14 + 1/16
x+1/4)^2 = 225/16
x + 1/4 = ± 15/4
x = -1/4 ± 15/4 = 7/2 or -4
Why did the quadratic equation go to the circus? It wanted to complete the "square" of entertainment! Alright, let's find the solution to this quadratic equation by completing the square.
First, let's rewrite the equation: 2x^2 + x - 28 = 0.
Now, we need to move the constant term to the other side: 2x^2 + x = 28.
To complete the square, we take half of the coefficient of x and square it. The coefficient of x is 1, so half of it is 1/2, and when we square it, we get 1/4.
Next, we add (1/4)^2 = 1/16 to both sides of the equation: 2x^2 + x + 1/16 = 28 + 1/16.
Simplifying this equation, we get: 2x^2 + x + 1/16 = 449/16.
Now, let's factor the left side of the equation as a perfect square: (x + 1/4)^2 = 449/16
Taking the square root of both sides, we have: x + 1/4 = ± √(449/16).
Now, to solve for x, we subtract 1/4 from both sides: x = -1/4 ± √(449/16).
Simplifying this further, we have two solutions: x = (-1 ± √449)/4. Voilà! The quadratic equation has revealed its circus performance with the solutions x = (-1 ± √449)/4.
To solve the quadratic equation 2x^2 + x - 28 = 0 by completing the square, follow the steps below:
Step 1: Make sure the coefficient of the x^2 term is 1. In this case, it is already 2, so we need to divide the entire equation by 2 to make it 1.
(x^2 + (1/2)x - 28/2) = 0
Simplified: (x^2 + (1/2)x - 14) = 0
Step 2: Move the constant term (-14) to the other side of the equation. This step is necessary to complete the square.
(x^2 + (1/2)x) = 14
Step 3: Take half of the coefficient of the x term (1/2), square it, and add it to both sides of the equation.
(x^2 + (1/2)x + (1/4)(1/4)) = 14 + (1/4)(1/4)
Simplified: (x + 1/4)^2 = 14 + 1/16
Step 4: Combine terms on the right side of the equation.
(x + 1/4)^2 = 225/16
Step 5: Take the square root of both sides of the equation to isolate x.
x + 1/4 = ± √(225/16)
Step 6: Simplify the square root on the right side.
x + 1/4 = ± 15/4
Step 7: Solve for x by subtracting 1/4 from both sides of the equation.
x = -1/4 ± 15/4
Step 8: Simplify the expression.
x = (-1 ± 15)/4
Step 9: Separate the two possible solutions.
x = (-1 + 15)/4 OR x = (-1 - 15)/4
Step 10: Simplify each solution.
x = 14/4 OR x = -16/4
Simplified: x = 7/2 OR x = -4
Therefore, the solutions to the quadratic equation 2x^2 + x - 28 = 0 by completing the square are x = 7/2 and x = -4.
To solve the equation 2x^2 + x - 28 = 0 by completing the square, follow these steps:
Step 1: Move the constant term to the right side of the equation:
2x^2 + x = 28
Step 2: Divide through by the coefficient of the squared term:
x^2 + (1/2)x = 14
Step 3: Take half of the coefficient of the x-term, square it, and add it to both sides of the equation:
x^2 + (1/2)x + [(1/2)(1/2)]^2 = 14 + [(1/2)(1/2)]^2
x^2 + (1/2)x + (1/16) = 14 + 1/16
Step 4: Simplify the equation:
x^2 + (1/2)x + (1/16) = 225/16
Step 5: Factor the left side of the equation:
(x + 1/4)^2 = 225/16
Step 6: Take the square root of both sides of the equation:
x + 1/4 = ±√(225/16)
Step 7: Simplify the square root on the right side:
x + 1/4 = ±15/4
Step 8: Solve for x:
x = -1/4 ± 15/4
Thus, the solutions to the equation 2x^2 + x - 28 = 0 by completing the square are:
x = (-1/4) + (15/4)
x = 7/2
and
x = (-1/4) - (15/4)
x = -16/4
x = -4