Different ways to solve quadratic equations
Questionsįactor each of the following polynomials and solve what you can. Checking for any others by using the discriminant reveals that all other solutions are complex or imaginary solutions. The two real solutions are x = 2 and x = -1. The factored (x^3 - 8) and (x^3 + 1) terms can be recognized as the difference of cubes. Now that the substituted values are factored out, replace the u with the original x^3. Here, it would be a lot easier if the expression for factoring was x^2 - 7x - 8 = 0.įirst, let u = x^3, which leaves the factor of u^2 - 7u - 8 = 0. This same strategy can be followed to solve similar large-powered trinomials and binomials.įactor the binomial x^6 - 7x^3 - 8 = 0. Factoring Method If the quadratic polynomial can be factored, the Zero Product Property may be used. There are four different methods used to solve equations of this type. In the following example, we will employ the above-mentioned quadratic formula. SOLVING QUADRATIC EQUATIONS A quadratic equation in is an equation that may be written in the standard quadratic form if. It utilized the coefficients a, b, and c for calculating the roots directory with this formula: x (-b ± (b2 - 4ac)) / 2a. Solving each of these terms yields the solutions x = \pm 3, \pm 2. The quadratic formula is a popular method for solving quadratic equations. This is done using the difference of squares equation: a^2 - b^2 = (a + b)(a - b).įactoring (x^2 - 9)(x^2 - 4) = 0 thus leaves (x - 3)(x + 3)(x - 2)(x + 2) = 0. To complete the factorization and find the solutions for x, then (x^2 - 9)(x^2 - 4) = 0 must be factored once more. Once the equation is factored, replace the substitutions with the original variables, which means that, since u = x^2, then (u - 9)(u - 4) = 0 becomes (x^2 - 9)(x^2 - 4) = 0.
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Often the easiest method of solving a quadratic equation is factoring. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics. Now substitute u for every x^2, the equation is transformed into u^2-13u+36=0. For example, equations such as (2x2 +3x10) and (x24 0) are quadratic equations. There is a standard strategy to achieve this through substitution.įirst, let u = x^2. Here, it would be a lot easier when factoring x^2 - 13x + 36 = 0.
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Solve for x in x^4 - 13x^2 + 36 = 0.įirst start by converting this trinomial into a form that is more common.