![]() Recall that we can rewrite an expression □ + 2 □ □ , with On the left-hand side of the equation, we can now follow the process for completing the □ □ from both sides of the equation, giving us □ + □ □ □ = − □ □. To begin isolating the □- and □ -terms, we subtract We can start our process of solving this equation byĭividing all the terms by □, which gives us □ + □ □ □ + □ □ = 0. Suppose we have a quadratic equation in the general form □ □ + □ □ + □ = 0, where □, □, and How To: Deriving the Quadratic Formula by Completing the Square We will now look at how we can use this process to derive a general formula for solving We have seen in this explainer how we can complete the square to solve a quadratic equation. Hence, our solution is □ = 1 3 + √ 7 3 or We can now add 1 3 to both sides to isolate □: Into our equation, □ − 2 3 □ = 2 3 , giving us Therefore, dividing 3 □ − 2 □ = 2 by 3, we have To begin solving an equation with a coefficient of □ that is notĮqual to 1, we transformthe equation by dividing everything by the value of this □ and continue to complete the square as we saw in the previousĮxample 6: Solving a Quadratic Equation by Completing the Squareīy completing the square, solve the equation 3 □ − 2 □ = 2 . □ > 1, we simply transform the equation by dividing all the terms by When we have a quadratic equation of the form □ □ + □ □ + □ = 0 , and Now we can add 1 4 to both sides to give us Into the equation □ − □ = 3 4 , giving us □-terms are on the same side of the equation. To begin solving this, we first need to rearrange the equation Where we need to rearrange an equation before solving using the completing the squareĮxample 5: Solving a Quadratic Equation by Completing the Squareīy completing the square, solve the equation In order to complete the square of a quadratic, we need to have the □ -Īnd □-terms on the same side of the equation. We can now add 11 to both sides, giving usįinally, adding 7 to both sides, we have □ = 7 ± √ 1 1. We can substitute □ − 1 4 □ = ( □ − 7 ) − 4 9 into the equation To begin, we put the equation □ − 1 4 □ + 3 8 = 0 into completed square Solve the equation □ − 1 4 □ + 3 8 = 0 by completing the square. ![]() Hence, we have two solutions: □ = − 4 + √ 2 6 andĮxample 4: Solving a Quadratic Equation by Completing the Square Subtract 4 from both sides to isolate □, giving us ![]() When taking the square root, we must consider both the positive and the negative values, Next, to solve the equation, we perform the same operation to both sides.Īdding 26 to both sides, ( □ + 4 ) = 2 6. We can now substitute □ + 8 □ = ( □ + 4 ) − 1 6 into the originalĮquation, □ + 8 □ − 1 0 = 0 , giving usĬollecting the constants, this will give us the completed square form On the other hand, we can expand the brackets of ( □ + 4 ): We have ( □ + 4 ) as our complete square. Given equation and take half of that value as our □ value. ( □ + □ ) , we take the coefficient of □ in our So, in order to create a perfect square of □ + 2 □ + □, we can see that we wouldĮquate the coefficients of the □-terms to get 2 □ = 8 Let us look at another example of using square roots to find the solution of an equation.Įxample 3: Solving a Quadratic Equation by Completing the Squareīy completing the square, solve the equation □ + 8 □ − 1 0 = 0 . It is important to remember that the use of ± indicates two solutions, so The positive and negative roots here as □ = ± √ 2 5. However, since − 5 × − 5 would also give an answer of 25, we can indicate If we wished to solve this for □, we would take the square root of both sides of theĮquation. To begin, let us consider the equation □ = 2 5. Completing the square can also give us useful information about the shape of the Of completing the square that we will cover in this explainer allows us to solve any quadraticĮquation. While factoring can be an efficient way to solveĪn equation, there are many quadratic equations that cannot be solved by factoring. Quadratic formula, or completing the square. We can solve equations in a number of different ways, including factoring, using the In this explainer, we will learn how to solve quadratics by completing the square.
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