Give all solutions of each nonlinear system of equations, including those with nonreal complex c

step 1 Now consider the given system of non -deny equations over here. The first equation is 5x square

Give all solutions of each nonlinear system of equations,...

Key Concepts

Nonlinear Systems

A nonlinear system involves two or more equations in which at least one equation is not linear, meaning it includes variables raised to a power other than one or products of variables. These systems typically require more advanced techniques than linear systems and can yield multiple or even complex solutions.

Complex Numbers

Complex numbers extend the real number system to include numbers of the form a + bi, where i is the square root of -1. They ensure that every polynomial equation has a solution (by the Fundamental Theorem of Algebra) and are crucial when solving equations that might not have real solutions.

Substitution Method

The substitution method is a common approach to solving systems of equations whereby one solves one of the equations for one variable in terms of the others and then substitutes that expression into the remaining equations. This reduces the number of variables and simplifies the system, eventually transforming it into a single equation in one variable.

Quadratic Equations

Quadratic equations, which are second-degree polynomials of the form ax² + bx + c = 0, frequently appear when solving nonlinear systems after substitution. Techniques such as factoring, completing the square, and the quadratic formula are used to find their roots, with the discriminant revealing whether the solutions are real or complex.

Transcript

00:01 Now consider the given system of non -deny equations over here.

00:04 The first equation is 5x square minus 2 y square is equal to 6.

00:10 And the second equation is given to us as xy is equal to 2.

00:14 So let's just mark this as equation 1 and this one as equation 2.

00:19 So now we are supposed to find out the solution for this given system over here.

00:24 So from equation 2, i can just find out the value of x in terms of y, like this.

00:30 So let's mark this as equation three now now i'm just going to substitute the value of x from equation three in equation one that means i'm using the substitution method over here so on doing that what i'll get simply five times two upon y whole square minus two y equal to 6.

00:52 If i just open up the brackets over here, so i'm going to get 20 upon y square minus 2 y square equal to 6.

01:02 Now just multiply both the sides by by square.

01:06 So we'll get 20 minus 2 y -res to the power 4 equal to 6 y square.

01:12 Now i'm just going to rearrange the terms over here.

01:16 So we'll get 2 y -reast -to -the -power 4 plus 6 y -square.

01:20 Minus 20 is equal to 0.

01:24 And if you want, you can just divide both the sides by 2.

01:28 So we are going to get 4 5 raise to the power 4 plus 3y square minus 10 is equal to 0.

01:36 Right.

01:36 Now, just assume y squared as t over here.

01:42 So when you do that, what you'll get? you'll get t squared plus 3t minus 10 is equal to 0.

01:49 Now, this is just a quadratic equation written in front of you.

01:54 And you need to just factorize this.

01:56 For which you can just use the factorization method or the quadratic formula...

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