Question

Use the method of completing the square and properties of radicals to derive the quadratic

formula for polynomial ax^2 + bx + c = 0

Answer 1

To derive the quadratic formula, start with the standard form ax^2 + bx + c = 0. Divide by a, rearrange, complete the square, then take the square root and solve for x, resulting in x = (-b ± √(b^2 - 4ac)) / (2a).

Step-by-step explanation:

To derive the quadratic formula using the method of completing the square, consider the general form of a quadratic equation:

ax^2 + bx + c = 0

First, divide the equation by a (provided that a is not equal to zero) to simplify:

x^2 + (b/a)x + (c/a) = 0

Next, move the constant term to the other side of the equation:

x^2 + (b/a)x = -c/a

To complete the square, add the square of half the coefficient of x to both sides:

x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2

Now, the left side of the equation is a perfect square trinomial. Factor it:

(x + b/2a)^2 = -c/a + (b^2/4a^2)

Then, take the square root of both sides:

x + b/2a = ±√(b^2 - 4ac) / (2a)

Finally, solve for x:

x = (-b ± √(b^2 - 4ac)) / (2a)

This is the quadratic formula, which gives the solutions for any quadratic equation of the form ax^2 + bx + c = 0.