Solve by completing the square



Completing the square is a mathematical technique that can be used to solve quadratic equations. The basic idea is to take an equation in the form of ax2 + bx + c = 0 and rewrite it as a perfect square. This can be done by adding a constant to both sides of the equation and then factoring the resulting trinomial.



Solving by completing the square

For example, consider the equation x2 + 6x + 9 = 0. To solve this equation by completing the square, we would first add a constant to both sides so that the left side becomes a perfect square: x2 + 6x + 9 + 4 = 4. Next, we would factor the trinomial on the left side to get (x + 3)2 = 4. Finally, we would take the square root of both sides to get x + 3 = ±2, which means that x = -3 ± 2 or x = 1 ± 2. In other words, the solutions to the original equation are x = -1, x = 3, and x = 5.

Solving by completing the square is a method that can be used to solve certain types of equations. The goal is to transform the equation into one that has a perfect square on one side, which can then be solved using the quadratic formula. This technique can be helpful when other methods, such as factoring, fail to provide a solution. To complete the square, start by taking the coefficient of the x^2 term and squaring it. This number will be added to both sides of the equation. Next, divide both sides of the equation by this number. The resulting equation should have a perfect square on one side. Finally, apply the quadratic formula to solve for x. With a little practice, solving by completing the square can be a helpful tool in solving equations.

Completing the square is a mathematical technique that can be used to solve equations and graph quadratic functions. The basic idea is to take an equation and rearrange it so that one side is a perfect square. For example, consider the equation x^2 + 6x + 9 = 0. This equation can be rewritten as (x^2 + 6x) + 9 = 0, which can then be simplified to (x+3)^2 = 0. From this, we can see that the solution is x = -3. Completing the square can also be used to graph quadratic functions. For example, the function y = x^2 + 6x + 9 can be rewritten as y = (x+3)^2 - 12. This shows that the function has a minimum value of -12 at x = -3. By completing the square, we can quickly and easily solve equations and graph quadratic functions.

Solving equations by completing the square is a useful technique that can be applied to a variety of equations. The first step is to determine whether the equation is in the form "x^2 + bx = c" or "ax^2 + bx = c." If the equation is in the latter form, it can be simplified by dividing everything by a. Once the equation is in the correct form, the next step is to add (b/2)^2 to both sides of the equation. This will complete the square on the left side of the equation. Finally, solve the resulting equation for x. This will give you the roots of the original equation. Solving by completing the square can be a little tricky, but with practice it can be a handy tool to have in your mathematical toolkit.

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