Atraeus is solving the quadratic equation by completing the square, an elegant technique that transforms complex equations into manageable forms, revealing the hidden solutions within. This method, steeped in mathematical finesse, empowers us to conquer quadratic equations with precision and clarity.

Completing the square involves a series of strategic steps, each meticulously designed to simplify the equation and expose its underlying structure. By adding and subtracting a carefully calculated value, we create a perfect square trinomial, a cornerstone of this technique.

Factoring this trinomial yields two binomials, paving the way for solving the equation and unveiling the elusive value of x.

Completing the Square to Solve Quadratic Equations

Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square trinomial. This transformation makes it possible to factor the trinomial and solve for the variable.

The steps involved in completing the square are as follows:

Steps for Completing the Square, Atraeus is solving the quadratic equation by completing the square

1. Add and subtract the square of half the coefficient of the x-term.

This step creates a perfect square trinomial.

The formula for calculating the value to be added and subtracted is: (coefficient of x-term /2)^2

2. Factor the perfect square trinomial.

This step involves factoring the trinomial into two binomials.

3. Solve for the variable xusing the factored binomials.

Use the square root property to isolate the variable.

Example Problems

Equation	Steps	Solution
x^2 + 4x + 3 = 0	Add and subtract (4/2)^2 = 4 Factor: (x+ 2)^2 1 = 0 Solve: x+ 2 = ±1, x= 1 ± 1	x = 2, 1
x^2 6x+ 5 = 0	Add and subtract (-6/2)^2 = 9 Factor: (x 3)^2 4 = 0 Solve: x 3 = ±2, x= 5, 1	x = 5, 1

Question Bank: Atraeus Is Solving The Quadratic Equation By Completing The Square

What is the essence of completing the square?

Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square trinomial, which can then be easily factored and solved.

Why is factoring the perfect square trinomial crucial?

Factoring the perfect square trinomial into two binomials allows us to apply the zero product property and solve for the variable x, revealing the solutions to the quadratic equation.