This quadratic equation could be solved by factoring, but well use the method of completing the square. The method is called solving quadratic equations by completing the square. #=> x^2 + b/ax + (b/(2a))^2 = -c/a + (b/(2a))^2 = b^2/(4a^2)-c/a# The method we shall study is based on perfect square trinomials and extraction of roots. Next, you want to get rid of the coefficient before x2 (a) because it won´t always be a perfect square. To do this, you will subtract 8 from both sides to get 3x2-6x15. Let's see what happens if we apply this to a general quadratic equation. To complete the square, first, you want to get the constant (c) on one side of the equation, and the variable (s) on the other side. The real trick here is observing in step 3 that the constant we need to add is equal to the square of half of the coefficient of #x#. ©Q D2x0o1S2P iKSuGtRa6 4S1oGf1twwuamrUei 0LjLoCM.W T PAMlcl4 drhisg2hatEsB XrqeQsger KvqeidM.2 v 5M1awdPeZ uwjirtbhi QIxnDftiFn4iOteeE qAwlXg1ezbor9aP u2B.w. Step 6: Solve the remaining linear equation: Create your own worksheets like this one with Infinite Algebra 2. Remember to account for both positive and negative roots. Step 5: Take the square root of both sides. Step 3: Add a constant to both sides which will allow us to factor the left hand side as #(x-h)^2#. If the problem had been an equation of: x2-44x 0 Completing the square would have resulted in x2-44x+484 484 (x-22)2 484 Take square root: x-22 +/- sqrt(484) Simplify: x 22 +/- 22 This results in: x22+22 44 And in x 0 Note: The equation would be easier to solve using factoring. Step 2: Add #2# to both sides to isolate the #x# terms. Step 1: Divide both sides by #2# to obtain #x^2# as the first term And we found these by completing the square.The idea behind completing the square is to add or subtract a constant to obtain the form #(x-h)^2# and then take a square root to be left with a linear equation. So therefore, the solutions to the equation □ squared minus 14□ plus 38 equals zero are □ is equal to seven plus root 11 or □ is equal to seven minus root 11. So then, we’re left with □ is equal to seven plus or minus root 11. And then if we take the square root of each side, we get □ minus seven is equal to plus or minus the square root of 11.Īnd now into our final stage, which is actually going to be let’s add seven to each side of the equation. So then if we simplify, we get □ minus seven all squared minus 11 is equal to zero.Īnd then our next stage is to actually add 11 to each side of the equation, which gives us □ minus seven all squared is equal to 11. If you square a negative, we get a positive. Add 21 to both sides of the equation: s 2 + 4 s 21 0. And we get that again because we had negative seven all squared. Move the constant to the right side of the equation and combine. So if you add a negative, it’s the same as just subtracting it. Well, negative 14 over two is negative seven. So we get □ minus seven all squared and it’s □ minus seven because we had □ plus and then negative 14 over two. Then we still have plus 38 is equal to zero. So if we actually apply this and complete the square of our first two terms, we’re gonna get □ plus and then we’ve got negative 14 over two because a coefficient of our □ is negative 14 and that’s all squared and then minus again negative 14 over two all squared. And these two terms are □ squared minus 14□. So if we look back to our equation, we can actually see that it’s the first two terms that are actually gonna apply to completing the square rule too. So the general rule if we have our expression in the form □ squared plus □□ is that this is equal to □ plus □ over two all squared - and that’s because we’ve actually halved the coefficient of □ - and then minus □ over two - again halving the coefficient of □ - and that is all squared. So what I’m gonna do is I’m going to recap that first. This question has asked us to solve the equation by completing the square. Solve the equation □ squared minus 14 □ plus 38 is equal to zero by completing the square.
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