Determine the roots of the quadratic equation by completing the square. Solve for exact solutions and approximate solutions. Check your solutions.
Y=x2-6x-12
x^2-6x - 12
(x^2-6x+9) - 12-9
(x-3)^2 - 21
so, setting
(x-3)^2 - 21 = 0
(x-3)^2 = 21
x-3 = ±√21
x = 3±√21
To determine the roots of the quadratic equation by completing the square, follow these steps:
Step 1: Write the given quadratic equation in the standard form, which is ax^2 + bx + c = 0. In this case, we have y = x^2 - 6x - 12.
Step 2: Move the constant term (-12) to the right side of the equation:
x^2 - 6x = 12
Step 3: To complete the square, take half of the x coefficient (-6), square it, and add it to both sides of the equation:
x^2 - 6x + (-6/2)^2 = 12 + (-6/2)^2
x^2 - 6x + 9 = 12 + 9
x^2 - 6x + 9 = 21
Step 4: Factor the perfect square trinomial on the left side:
(x - 3)^2 = 21
Step 5: Take the square root of both sides:
√((x - 3)^2) = ±√21
x - 3 = ±√21
Step 6: Solve for x by adding 3 to both sides of the equation:
x = 3 ± √21
Therefore, the exact solutions of the quadratic equation y = x^2 - 6x - 12 are x = 3 + √21 and x = 3 - √21.
To find the approximate solutions, you can use a calculator or a computer program. The approximate solutions to a high degree of precision are x ≈ 6.582 and x ≈ -0.582.
To check the solutions, substitute each value of x back into the original equation y = x^2 - 6x - 12 and see if both sides of the equation are equal. If they are equal, then the given values of x are indeed the roots of the equation.