Equations reducible to quadratics are algebraic equations that can be transformed into quadratic equations through a series of algebraic manipulations. These equations are often easier to solve than more complex equations, as the techniques for solving quadratic equations are well-known and well-established.
One way to recognize an equation that is reducible to a quadratic is to look for a pattern in the terms. For example, if an equation has terms that are perfect squares, such as x^2 or (y+2)^2, it may be reducible to a quadratic. Similarly, if an equation has terms that are the product of two variables, such as xy or 2xz, it may also be reducible to a quadratic.
To reduce an equation to a quadratic, one must first use algebraic techniques to isolate the quadratic term. This can involve combining like terms, factoring, or using the distributive property. Once the quadratic term has been isolated, it can be rewritten in the standard form of a quadratic equation, which is ax^2 + bx + c = 0.
Once the equation has been rewritten in this form, it can be solved using the quadratic formula, which is:
x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
The quadratic formula allows us to find the roots of the quadratic equation, which are the values of x that make the equation true. These roots can then be plugged back into the original equation to verify that they are indeed solutions.
While equations reducible to quadratics may seem complex at first, the techniques for solving them are actually quite simple and straightforward. By recognizing the patterns in the terms of the equation and using the quadratic formula, we can quickly and easily find the solutions to these types of equations.
Equations Reducible to the Quadratic Equation
Note as well that all we really needed to notice here is that the exponent on the first term was twice the exponent on the second term. A: Some of the equations of various types can be reduced to quadratic form. One way to think of this is as follows: Let Then we have , substitute into to get, Notice that the change in variable from to has resulted in a quadratic equation that can be easily factored due to the fact that it is a square of a simple binomial: The solution for is, Because we go back to the variable , Therefore, the roots of the factor are, The other root of is since the function clearly equals when. Substituting for , we get , in which case , or in which case. They are advanced to linear functions and give a significant move away from attachment to straight lines. Dividing the equation by both sides in the preferable application of applying square by both sides can make the square complete. Explanation: can be rewritten in quadratic form by setting , and, consequently, ; the resulting equation is as follows: By the reverse-FOIL method we can factor the trinomial at left.
In most cases, check the exponents of the given equation. Method: In this type of equation, we group two terms and expand to have a similarity in the terms. Roots of the quadratic equation can be referred to as the most useful and simple formula to get the reducible value. Further Discussion The change of variable was a tool we used to write the quadratic factor in a more familiar form, but we could have just factored the original function in terms of as follows, Setting this to zero gives the same solution set, Explanation: can be rewritten in quadratic form by setting , and, consequently, ; the resulting equation is as follows: By the reverse-FOIL method we can factor the trinomial at left. Let us learn the different types of equations that can be reduced to quadratic form with solved examples. That need not always be the case however. A quadratic polynomial, when equated to zero, becomes a quadratic equation.
