Solving quadratic equations4/9/2024 For example, expand the factored expression ( x − 2 ) ( x + 3 ) ( x − 2 ) ( x + 3 ) by multiplying the two factors together.Ĭalculate the discriminant b 2 − 4 a c b 2 − 4 a c for each equation and state the expected type of solutions.ī 2 − 4 a c = ( 4 ) 2 − 4 ( 1 ) ( 4 ) = 0. So, in that sense, the operation of multiplication undoes the operation of factoring. Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero. Solving by factoring depends on the zero-product property, which states that if a ⋅ b = 0, a ⋅ b = 0, then a = 0 a = 0 or b = 0, b = 0, where a and b are real numbers or algebraic expressions. If a quadratic equation can be factored, it is written as a product of linear terms. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation. 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. For example, equations such as 2 x 2 + 3 x − 1 = 0 2 x 2 + 3 x − 1 = 0 and x 2 − 4 = 0 x 2 − 4 = 0 are quadratic equations. Solving Quadratic Equations by FactoringĪn equation containing a second-degree polynomial is called a quadratic equation. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods. Proportionally, the monitors appear very similar. No such general formulas exist for higher degrees.The computer monitor on the left in Figure 1 is a 23.6-inch model and the one on the right is a 27-inch model. So in conclusion, there are only general formulae for 1st, 2nd, 3rd, and 4th degree polynomials. It's that we will never find such formulae because they simply don't exist. So it's not that we haven't yet found a formula for a degree 5 or higher polynomial. The Abel-Ruffini Theorem establishes that no general formula exists for polynomials of degree 5 or higher. In fact, the highest degree polynomial that we can find a general formula for is 4 (the quartic). Both of these formulas are significantly more complicated and difficult to derive than the 2nd degree quadratic formula! Here is a picture of the full quartic formula:īe sure to scroll down and to the right to see the full formula! It's huge! In practice, there are other more efficient methods that we can employ to solve cubics and quartics that are simpler than plugging in the coefficients into the general formulae. These are the cubic and quartic formulas. There are general formulas for 3rd degree and 4th degree polynomials as well. Similar to how a second degree polynomial is called a quadratic polynomial. A third degree polynomial is called a cubic polynomial. A trinomial is a polynomial with 3 terms. First note, a "trinomial" is not necessarily a third degree polynomial.
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