solve quadratic equation by completing the square

Let's take a look at the part containing the unknown x x x we have x 2 + 6 x x^2 + 6x x 2 + 6 x. Solve using the completing the square method: x 2 + 6 x + 5 = 0 x^2 + 6x + 5 = 0 x 2 + 6 x + 5 = 0. In fact, in this example we didn't have to complete the square, because the perfect square trinomial was already there, staring at us defiantly!Įxample 2. Thus, our problem can be rewritten as ( x + 2 ) 2 = 0 (x+2)^2 = 0 ( x + 2 ) 2 = 0. We immediately recognize the short multiplication formula working in reverse: ( x + 2 ) 2 = x 2 + 4 x + 4 (x+2)^2 =x^2 + 4x + 4 ( x + 2 ) 2 = x 2 + 4 x + 4. Solve by completing the square: x 2 + 4 x + 4 = 0 x^2 + 4x + 4 = 0 x 2 + 4 x + 4 = 0. Let's discuss a few examples of solving quadratic equations by completing the square.Įxample 1. If b 2 / 4 If b 2 / 4 = c b^2/4 = c b 2 /4 = c, then we have one solution, and it is equal to − b / 2 -b/2 − b /2.If b 2 / 4 > c b^2/4 > c b 2 /4 > c, then there's no solution.The next step depends on the sign of the right-hand side: We have to perform the same operation on the right side! Finally, our equation is equivalent to ( x + b / 2 ) 2 = − c + b 2 / 4 (x+b/2)^2 = -c + b^2/4 ( x + b /2 ) 2 = − c + b 2 /4. To get from c c c to b 2 / 4 b^2/4 b 2 /4, we have to subtract c c c and add b 2 / 4 b^2/4 b 2 /4. The first two terms are the same, but the last terms differ. If the leading coefficient of your quadratic equation is not 1 1 1 (i.e., if the polynomial is not monic), then divide both sides by a a a.Īssume we have the expression x 2 + b x + c = 0 x^2 + bx + c = 0 x 2 + b x + c = 0. When rewriting in perfect square format the value in the parentheses is the x-coefficient of the parenthetical expression divided by 2 as found in Step 4.In order to solve a quadratic equation by completing the square, follow these steps: So this value must also be added to the right-hand side.

When adding the result to the parenthetical expression on the left-hand side the total value added is a x result. To keep the equation true what is done to one side must also be done to the other. This changes the value of the x-coefficient. When rewriting in perfect square format the value in the parentheses is the x-coefficient of the parenthetical expression divided by 2, as found in Step 4.

Remember that when taking the square root on the right-hand side the answer can be positive or negative.Įxpression on the left-hand side the total value added is Take the square root of both sides of the equation. Now that the square has been completed, solve for x. When rewriting in perfect square format the value in the parentheses is the x-coefficient of the parenthetical expression divided by Rewrite the left-hand side as a perfect square and simplify the right-hand side. When adding the result to the parentheticalĮxpression on the left-hand side the total value added is a x result.

This changes the value of the x -coefficient.Ĭomplete the square of the expression in parentheses on the left-hand side of the equation.ĭivide the x-coefficient by two and square the result.Īdd the result from Step 4 to the parenthetical expression on the left-hand side. Move c, the constant term, to the right-hand side of the equation. This equation is already in the proper form where a = 2 and c = -5.

Solve quadratic equation by completing the square