We set each factor equal to 0, leading to If the product of three factors is equal to 0, at least one of the individual factors must be equal to 0. The fully factored form of the equation is therefore Is the difference of two squares and hence can be factored as follows: We observe that the quadratic expression □ − 9 We now have the product of a linear term and a quadratic expression. The equation can therefore be factored as We observe that the two terms on the left-hand side of the equation have a common factor of 2 □. This equation is of degree 3 and hence is a cubic equation. The equation we are given is not a quadratic equation. We summarize the steps involved in solving a quadratic equation by factoring in the process below.Įxample 3: Solving an Equation by First Identifying the Highest Common Factor between the Termsįind the solution set for 2 □ = 1 8 □ . We can check our answer by substituting □ = 4 back into the original quadratic equation: This is a repeated root, but we only need to list it once as our solution. We have a repeated factor of ( □ − 4 ), so we only have one equation to solve: 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 A quadratic equation is any equation that can be expressed in the form □ □ + □ □ + □ = 0 , where □, □,
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