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How many solutions does this system of equations have?

[tex]\ \textless \ br/\ \textgreater \ \begin{array}{l}\ \textless \ br/\ \textgreater \ y = 2x + 5 \\\ \textless \ br/\ \textgreater \ y = x^3 + 4x^2 + x + 2\ \textless \ br/\ \textgreater \ \end{array}\ \textless \ br/\ \textgreater \ [/tex]

A. no solution
B. 1 solution
C. 2 solutions
D. 3 solutions



Answer :

GinnyAnswer
To determine how many solutions the system of equations has, we need to find the points of intersection between the two equations.

The given system of equations is:
[tex]\[ \begin{array}{l} y = 2x + 5 \\ y = x^3 + 4x^2 + x + 2 \end{array} \][/tex]

Since both expressions are set equal to [tex]\( y \)[/tex], we can set the right sides of the equations equal to each other:
[tex]\[ 2x + 5 = x^3 + 4x^2 + x + 2 \][/tex]

To find the solutions, we need to solve this equation:
[tex]\[ 2x + 5 = x^3 + 4x^2 + x + 2 \][/tex]

First, bring all terms to one side of the equation:
[tex]\[ 0 = x^3 + 4x^2 + x + 2 - 2x - 5 \][/tex]

Simplify the equation:
[tex]\[ 0 = x^3 + 4x^2 - x - 3 \][/tex]

Now, we need to find the roots of the polynomial equation [tex]\( x^3 + 4x^2 - x - 3 = 0 \)[/tex]. The number of roots of a cubic equation corresponds to the number of solutions for [tex]\( x \)[/tex] where the two equations intersect.

This specific polynomial has exactly 3 roots. Therefore, the equation has 3 solutions.

So, the number of solutions to the system of equations is:

[tex]\[ \boxed{3} \][/tex]

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