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

Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
y = x^2 + 6x + 9 
x + 2y = -2

Accepted Answer

Hey, everyone. Here we are asked to provide all solutions for the given nonlinear system of equations including non-real complex components. Here we have two given equations. The first equation is Y is equal to X squared plus 12 X plus 36. And the second given equation is two X plus Y is equal to negative four. Here we have four answer choice options. A through D, each of them containing a solution set with two ordered pairs. So here we see from our given equations that our first equation is already solved for Y. So here we can utilize the substitution method by substituting in our first equation into our second equation and starting by solving for X. So here again, we have our first equation and if we recall our second equation, we have two X plus Y is equal to negative four. While we will now have two X plus our first equation which is X squared plus 12 X plus 36 this is still equal to negative four. Now, we just want to combine like terms on the left side of our equation. So we have X squared plus 14 X plus 36 is equal to negative four. Now, just making the right hand side of our equation zero, we want to just add four to both sides of the equation. So we have X squared plus 14 X plus 40 is equal to zero. Now, we just want to factor the left side of the equation. So we have the quantity of X plus 10 times the quantity of X plus four is equal to zero. Now, we can recall the zero factor property which allows us to see each individual set of parentheses equal to zero to solve four values of X. So starting with our first set of parentheses, we have the expression X plus 10. And again, we can set this equal to zero. And now solving for X, we just subtract 10 from both sides of the equation. So we have X is equal to negative 10. Now repeating this with the other set of parentheses, we have the expression X plus four which again we set equal to zero. Now solving for X, we just subtract four from both sides of the equation. And so we have our other X value is negative four. So now with these two values of X that we found, we need to find our values of Y. And so to do this, we just need to substitute in these values of X into our first equation and solve for Y. So we can begin with our first found value of X which is negative 10. And again, we substitute this value into our first equation. So we have Y is equal to the quantity of negative 10 squared plus 12 times negative 10 plus 36. Now just simplifying our equation, we have Y is equal to 100 minus 120 plus 36. Now combining all of our terms, we have Y is equal to 16. So here we have found that when X is equal to negative 10, that Y is equal to 16. So now we just want to repeat this process with the other value of X that we found, which is negative four. So again, substituting in this value into our first equation, we have Y is equal to the quantity of negative four squared plus 12 times negative four plus 36. Now just simplifying our equation, we have Y is equal to 16 minus 48 plus 36. And simplifying our terms, we find that Y is equal to four. So here we have also found that when X is equal to negative four, that Y is equal to positive four. So with all of these solutions, we can summarize our findings by writing these solutions in a solution set containing two ordered pairs. So our first ordered pair, we have negative 10 and 16. And from our second set of solutions, we have a second ordered pair of negative four and four. So with our solutions, we are left here with answer choice D where again, we have the solution set of two ordered pairs, one of which is negative 10 and 16 and the other ordered pair is negative four and four. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. y = x^2 + 6x + 9 x + 2y = -2

Key Concepts

Nonlinear Equations

Substitution Method

Complex Solutions

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