Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting them in the form of a perfect square trinomial, which makes it easier to solve for the variable.

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, such as $$(a + b)^2 = a^2 + 2ab + b^2$$ .

1. Move the constant to the other side: $$p^2 + 2p = 63$$ . 2. Take half of the coefficient of $$p$$ (which is 1), square it (1), and add it to both sides: $$p^2 + 2p + 1 = 64$$ . 3. Factor: $$(p + 1)^2 = 64$$ . 4. Solve: $$p + 1 = \text{±}8$$ , giving $$p = 7, -9$$ .

To complete the square, take half of -12 (which is -6), square it (36), so $$c = 36$$ . The expression becomes $$(b - 6)^2$$ .

1. Rearrange: $$m^2 + 2m = 42$$ . 2. Add 1 to both sides: $$m^2 + 2m + 1 = 43$$ . 3. Factor: $$(m + 1)^2 = 43$$ . 4. Solve: $$m + 1 = \text{±}\sqrt{43}$$ , giving approximate values of $$m = 5.55, -7.55$$ .

The formula for completing the square for a quadratic equation $$ax^2 + bx + c$$ is: 1. Divide by $$a$$ (if @@a eq 1 $$). 2. Rewrite as$$ a(x + rac{b}{2a})^2 - rac{b^2 - 4ac}{4a}@@.

The term $$b/2$$ is crucial because it helps to form the perfect square trinomial, allowing for easier factoring and solving of the quadratic equation.

To complete the square, take half of 14 (which is 7), square it (49), so $$c = 49$$ . The expression becomes $$(p + 7)^2$$ .

1. Recognize that $$x^2 + 6x + 9 = (x + 3)^2$$ . 2. Set $$(x + 3)^2 = 0$$ . 3. Solve: $$x + 3 = 0$$ , giving $$x = -3$$ .

Completing the square is a method to derive the quadratic formula. It transforms the standard form of a quadratic equation into a solvable format.