Now we use our algebra skills to solve for "x". I have a lesson on the Quadratic Formula, which provides worked examples and shows the connection between the discriminant (the 'b 2 4ac' part inside the square root), the number and type of solutions of the quadratic equation, and the graph of the related parabola. Total time = time upstream + time downstream = 3 hours (to travel 8 km at 4 km/h takes 8/4 = 2 hours, right?) We can turn those speeds into times using: when going downstream, v = x+2 (its speed is increased by 2 km/h).when going upstream, v = x−2 (its speed is reduced by 2 km/h).Let v = the speed relative to the land (km/h)īecause the river flows downstream at 2 km/h:.Let x = the boat's speed in the water (km/h).There are two speeds to think about: the speed the boat makes in the water, and the speed relative to the land: What is the boat's speed and how long was the upstream journey? The negative value of x make no sense, so the answer is:Įxample: River Cruise A 3 hour river cruise goes 15 km upstream and then back again. The desired area of 28 is shown as a horizontal line. There are many ways to solve it, here we will factor it using the "Find two numbers that multiply to give ac, and add to give b" method in Factoring Quadratics: 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): These are the four general methods by which we can solve a quadratic equation. (Note: t is time in seconds) The height starts at 3 m: Answer: There are various methods by which you can solve a quadratic equation such as: factorization, completing the square, quadratic formula, and graphing. The sides of the deck are 8, 15, and 17 feet.Ignoring air resistance, we can work out its height by adding up these three things: ![]() ![]() Since \(x\) is a side of the triangle, \(x=−8\) does not It is a quadratic equation, so get zero on one side. Since this is a right triangle we can use the It emphasizes the importance of understanding the Zero Product Property, which states that if a product is equal to zero, at least one of the factors must also be zero. We are looking for the lengths of the sides This lesson delves into the method of solving quadratic equations by factoring. Find the lengths of the sides of the deck. The length of one side will be 7 feet less than the length of the other side. Justine wants to put a deck in the corner of her backyard in the shape of a right triangle, as shown below. \(W=−5\) cannot be the width, since it's negative. The first thing I realize in this problem is that one side of the equation doesn’t contain zero. Example 5: Solve the quadratic equation below using the Factoring Method. Use the formula for the area of a rectangle. You should back-substitute to verify that latexx 0 /latex, latexx ,3 /latex, and latexx 3 /latex are the correct solutions. ![]() The area of the rectangular garden is 15 square feet. Restate the important information in a sentence. In problems involving geometric figures, a sketch can help you visualize the situation. The length of the garden is two feet more than the width. \)Ī rectangular garden has an area of 15 square feet.
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