Solving quadratic equations by completing the square.
Could you show me every step how to do it. That way I'll know probably know how to do the other ones. With that method you use.
1/2x^2 + x -13 = 0
yes
To solve the quadratic equation 1/2x^2 + x -13 = 0 by completing the square, follow these steps:
Step 1: Set the equation equal to zero.
1/2x^2 + x - 13 = 0
Step 2: Rearrange the equation so that the constant term is on the other side of the equation.
1/2x^2 + x = 13
Step 3: Divide the entire equation by the coefficient of x^2 to make the coefficient 1.
(1/2)x^2 + x = 13
Step 4: Move the constant term to the other side of the equation.
(1/2)x^2 + x + (-13) = 0
Step 5: Next, you need to determine the value you'll add and subtract from the equation to complete the square. To do this, take the coefficient of x, divide it by 2, and then square the result. In this case, the coefficient of x is 1, so:
(1/2) * (1/2) = 1/4
Step 6: Add and subtract the value determined in Step 5 to the left side of the equation.
(1/2)x^2 + x + (1/4) - (1/4) - 13 = 0
Step 7: Rearrange the equation so that the terms with x are grouped together.
(1/2)x^2 + x + (1/4) = (1/4) + 13
Step 8: Simplify both sides of the equation.
(1/2)x^2 + x + (1/4) = 53/4
Step 9: The left side of the equation can now be written as a perfect square trinomial. To factor it, take half of the coefficient of x and square it. Then, add that value inside the parentheses.
(1/2)(x + 1/2)^2 = 53/4
Step 10: To solve for x, take the square root of both sides of the equation.
√[(1/2)(x + 1/2)^2] = ±√(53/4)
Step 11: Simplify both sides of the equation.
(x + 1/2) = ±√(53/4)
Step 12: Subtract 1/2 from both sides of the equation.
x = -1/2 ±√(53/4)
Therefore, the solutions to the quadratic equation 1/2x^2 + x - 13 = 0, obtained by completing the square, are:
x = -1/2 + √(53/4) and x = -1/2 - √(53/4).