You have designed a new style of sports bicycle! So the ball reaches the highest point of 12.8 meters after 1.4 seconds. Then find the height using that value (1.4) The method is explained in Graphing Quadratic Equations, and has two steps:įind where (along the horizontal axis) the top occurs using −b/2a: Note: You can find exactly where the top point is! The factors of −15 are: −15, −5, −3, −1, 1, 3, 5, 15īy trying a few combinations we find that −15 and 1 work Multiply to give a×c, and add to give b" method in Factoring Quadratics: There are many ways to solve it, here we will factor it using the "Find two numbers that It looks even better when we multiply all terms by −1: (Note for the enthusiastic: the -5t 2 is simplified from -(½)at 2 with a=9.8 m/s 2)Īdd them up and the height h at any time t is:Īnd the ball will hit the ground when the height is zero: Gravity pulls it down, changing its position by about 5 m per second squared: It travels upwards at 14 meters per second (14 m/s): (Note: t is time in seconds) The height starts at 3 m: Furthermore, equations often have complex solutions.Ignoring air resistance, we can work out its height by adding up these three things: However, not all quadratic equations will factor. If an equation factors, we can solve it by factoring. If this is the case, then the original equation will factor. Note: In the previous example the solutions are integers. x + 1 = ± 49 x + 1 = ± 7 x = − 1 ± 7Īt this point, separate the “plus or minus” into two equations and solve each individually. X 2 + 2 x = 48 C o m p l e t e t h e s q u a r e. To complete the square, add 1 to both sides, complete the square, and then solve by extracting the roots. Next, find the value that completes the square using b = 2. Solve by completing the square: x 2 + 2 x − 48 = 0. This method allows us to solve equations that do not factor. for any real number k,Īpplying the square root property as a means of solving a quadratic equation is called extracting the root Applying the square root property as a means of solving a quadratic equation. In general, this describes the square root property For any real number k, if x 2 = k, then x = ± k. Here we see that x = ± 3 2 are solutions to the resulting equation. If we take the square root of both sides of this equation, we obtain the following: The equation 4 x 2 − 9 = 0 is in this form and can be solved by first isolating x 2. The goal in this section is to develop an alternative method that can be used to easily solve equations where b = 0, giving the form Here we use ± to write the two solutions in a more compact form. For example, we can solve 4 x 2 − 9 = 0 by factoring as follows:Ĥ x 2 − 9 = 0 ( 2 x + 3 ) ( 2 x − 3 ) = 0 2 x + 3 = 0 or 2 x − 3 = 0 2 x = − 3 2 x = 3 x = − 3 2 x = 3 2 If the quadratic expression factors, then we can solve the equation by factoring. Quadratic equations can have two real solutions, one real solution, or no real solution-in which case there will be two complex solutions. A solution to such an equation is a root of the quadratic function defined by f ( x ) = a x 2 + b x + c. Where a, b, and c are real numbers and a ≠ 0. Recall that a quadratic equation is in standard form Any quadratic equation in the form a x 2 + b x + c = 0, where a, b, and c are real numbers and a ≠ 0.
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