Question

the first step in determining the solution to the system of equations, $y = -x^2 - 4x - 3$ and $y = 2x + 5$, algebraically is to set the two equations equal as $-x^2 - 4x - 3 = 2x + 5$. what is the next step?
\bigcirc set $y = 0$ in $y = -x^2 - 4x - 3$.
\bigcirc factor each side of the equation.
\bigcirc use substitution to create a one - variable equation.
\bigcirc combine like terms onto one side of the equation.

Explanation:

To solve the system \( -x^{2}-4x - 3=2x + 5 \), after setting the equations equal, the next step is to combine like terms on one side to form a quadratic equation in standard form (\(ax^{2}+bx + c = 0\)). Let's analyze the options:

• Option 1: Setting \(y = 0\) in \(y=-x^{2}-4x - 3\) is not relevant here as we already equated the two \(y\)-expressions.
• Option 2: Factoring each side is not the next step since the right - hand side \(2x + 5\) is linear and not factorable in a way that helps here, and the left - hand side is a quadratic. We first need to get all terms on one side.
• Option 3: Substitution was already used when we set \(-x^{2}-4x - 3=2x + 5\) (we substituted \(y\) from one equation into the other).
• Option 4: We need to move all terms to one side (combine like terms) to get the equation in the form \(ax^{2}+bx + c = 0\). For example, add \(x^{2}+4x + 3\) to both sides and subtract \(2x + 5\) from both sides: \(0=x^{2}+6x + 8\) (or \(-x^{2}-6x - 8 = 0\)).

Answer:

D. Combine like terms onto one side of the equation.