Class 10 > Quadratic Equations

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Page 1 Solving Equations Page 2 Solving Equations A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a ? 0 0 2 ? ? ? c bx ax To solve a quadratic equation we get it in the form above and see if it will factor. 6 5 2 ? ? x x Get form above by subtracting 5x and adding 6 to both sides to get 0 on right side. -5x + 6 -5x + 6 0 6 5 2 ? ? ? x x Factor. ? ? ? ? 0 2 3 ? ? ? x x Use the Null Factor law and set each factor = 0 and solve. 0 2 or 0 3 ? ? ? ? x x 3 ? x 2 ? x So if we have an equation in x and the highest power is 2, it is quadratic. Page 3 Solving Equations A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a ? 0 0 2 ? ? ? c bx ax To solve a quadratic equation we get it in the form above and see if it will factor. 6 5 2 ? ? x x Get form above by subtracting 5x and adding 6 to both sides to get 0 on right side. -5x + 6 -5x + 6 0 6 5 2 ? ? ? x x Factor. ? ? ? ? 0 2 3 ? ? ? x x Use the Null Factor law and set each factor = 0 and solve. 0 2 or 0 3 ? ? ? ? x x 3 ? x 2 ? x So if we have an equation in x and the highest power is 2, it is quadratic. In this form we could have the case where b = 0. 0 2 ? ? ? c bx ax Remember standard form for a quadratic equation is: 0 2 ? ?c ax 0 0 2 ? ? ? c x ax When this is the case, we get the x 2 alone and then square root both sides. 0 6 2 2 ? ? x Get x 2 alone by adding 6 to both sides and then dividing both sides by 2 + 6 + 6 6 2 2 ? x 2 2 3 2 ? x Now take the square root of both sides remembering that you must consider both the positive and negative root. ? 3 ? ? x Let's check: ? ? 0 6 3 2 2 ? ? ? ? 0 6 3 2 2 ? ? ? 0 6 6 ? ? 0 6 6 ? ? Now take the square root of both sides remembering that you must consider both the positive and negative root. Page 4 Solving Equations A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a ? 0 0 2 ? ? ? c bx ax To solve a quadratic equation we get it in the form above and see if it will factor. 6 5 2 ? ? x x Get form above by subtracting 5x and adding 6 to both sides to get 0 on right side. -5x + 6 -5x + 6 0 6 5 2 ? ? ? x x Factor. ? ? ? ? 0 2 3 ? ? ? x x Use the Null Factor law and set each factor = 0 and solve. 0 2 or 0 3 ? ? ? ? x x 3 ? x 2 ? x So if we have an equation in x and the highest power is 2, it is quadratic. In this form we could have the case where b = 0. 0 2 ? ? ? c bx ax Remember standard form for a quadratic equation is: 0 2 ? ?c ax 0 0 2 ? ? ? c x ax When this is the case, we get the x 2 alone and then square root both sides. 0 6 2 2 ? ? x Get x 2 alone by adding 6 to both sides and then dividing both sides by 2 + 6 + 6 6 2 2 ? x 2 2 3 2 ? x Now take the square root of both sides remembering that you must consider both the positive and negative root. ? 3 ? ? x Let's check: ? ? 0 6 3 2 2 ? ? ? ? 0 6 3 2 2 ? ? ? 0 6 6 ? ? 0 6 6 ? ? Now take the square root of both sides remembering that you must consider both the positive and negative root. 0 2 ? ? ? c bx ax What if in standard form, c = 0? 0 0 2 ? ? ?bx ax We could factor by pulling an x out of each term. 0 3 2 2 ? ? x x Factor out the common x ? ? 0 3 2 ? ? x x Use the Null Factor law and set each factor = 0 and solve. 0 3 2 or 0 ? ? ? x x 2 3 or 0 ? ? x x If you put either of these values in for x in the original equation you can see it makes a true statement. Page 5 Solving Equations A quadratic equation is an equation equivalent to one of the form Where a, b, and c are real numbers and a ? 0 0 2 ? ? ? c bx ax To solve a quadratic equation we get it in the form above and see if it will factor. 6 5 2 ? ? x x Get form above by subtracting 5x and adding 6 to both sides to get 0 on right side. -5x + 6 -5x + 6 0 6 5 2 ? ? ? x x Factor. ? ? ? ? 0 2 3 ? ? ? x x Use the Null Factor law and set each factor = 0 and solve. 0 2 or 0 3 ? ? ? ? x x 3 ? x 2 ? x So if we have an equation in x and the highest power is 2, it is quadratic. In this form we could have the case where b = 0. 0 2 ? ? ? c bx ax Remember standard form for a quadratic equation is: 0 2 ? ?c ax 0 0 2 ? ? ? c x ax When this is the case, we get the x 2 alone and then square root both sides. 0 6 2 2 ? ? x Get x 2 alone by adding 6 to both sides and then dividing both sides by 2 + 6 + 6 6 2 2 ? x 2 2 3 2 ? x Now take the square root of both sides remembering that you must consider both the positive and negative root. ? 3 ? ? x Let's check: ? ? 0 6 3 2 2 ? ? ? ? 0 6 3 2 2 ? ? ? 0 6 6 ? ? 0 6 6 ? ? Now take the square root of both sides remembering that you must consider both the positive and negative root. 0 2 ? ? ? c bx ax What if in standard form, c = 0? 0 0 2 ? ? ?bx ax We could factor by pulling an x out of each term. 0 3 2 2 ? ? x x Factor out the common x ? ? 0 3 2 ? ? x x Use the Null Factor law and set each factor = 0 and solve. 0 3 2 or 0 ? ? ? x x 2 3 or 0 ? ? x x If you put either of these values in for x in the original equation you can see it makes a true statement. 0 2 ? ? ? c bx ax What are we going to do if we have non-zero values for a, b and c but can't factor the left hand side? 0 3 6 2 ? ? ? x x This will not factor so we will complete the square and apply the square root method. First get the constant term on the other side by subtracting 3 from both sides. 3 6 2 ? ? ? x x ___ 3 ___ 6 2 ? ? ? ? ? x x We are now going to add a number to the left side so it will factor into a perfect square. This means that it will factor into two identical factors. If we add a number to one side of the equation, we need to add it to the other to keep the equation true. Let's add 9. Right now we'll see that it works and then we'll look at how to find it. 9 9 6 9 6 2 ? ? ? x xRead More

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