![]() ![]() To solve a quadratic equation in its factored form, we set each factor equal to 0 and then solve the resulting linear equations. The quadratic is, therefore, a perfect square. 16 is equal to 4 ,Īdditionally, we observe that the middle term, − 8 □, is equal to the negative of twice the square root of the first term multiplied Upon inspection, we observe that both the first and third terms in this quadratic equation are perfect squares. Solve the equation □ − 8 □ + 1 6 = 0 by factoring. In the next example, we will demonstrate how to solve a quadratic equation by first recognizing it as a perfect square and hence writing the quadratic in its factored form.Įxample 2: Solving a Simple Quadratic Equation by Factoring This can be expressed as the solution set − 4 3, 3 2 . The solutions to the quadratic equation ( 2 □ − 3 ) ( 3 □ + 4 ) = 0 are □ = 3 2 and The second equation can be solved by subtracting 4 from each side and then dividing by 3: The first equation can be solved by adding 3 to each side and then dividing by 2: ![]() ![]() We therefore have two linear equations to solve. The only way the product of these two expressions can be 0 is if one of the factors individually is equal to 0. We are given that the product of the two linear expressions ( 2 □ − 3 )Īnd ( 3 □ + 4 ) is 0. This quadratic equation is already in a factored form. In our first example, we will demonstrate the process of solving a quadratic equation given its factored form.Įxample 1: Solving a Prefactored Equation It is important to remember that not every quadratic equation is factorable, so the methods we discuss here can only be applied to those that are. In some cases, the two solutions may coincide to give one repeated root, in which case we only give this value once as the solution. There are therefore two solutions, or roots, to the given quadratic equation: □ = − □ □ and □ = − □ □. So, to find all solutions to the given equation, we set each factor equal to 0, leading to two linear equations: The key to solving such an equation is to recognize that if the product of two (or more) factors is equal to 0, then at least one of the individual factors Suppose we have a quadratic equation in its factored form, The focus of this explainer is the application of these skills to solving quadratic equations. recognizing a quadratic as the difference of two squares,.recognizing a quadratic as a perfect square,.We should already be familiar with a number of methods for factoring quadratic expressions, including The sides of the deck are 8, 15, and 17 feet.A quadratic equation is any equation that can be expressed in the form □ □ + □ □ + □ = 0 , where □, □, 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 We are looking for the lengths of the sides 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. Use the formula for the area of a rectangle. 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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