In Exercises 29–42, solve each system by the method of your choice. {y=(x+3)2x+2y=−2\begin{cases} y = (x + $$3)^2$$ \\ x + 2y = -2 \end{cases}{y=(x+3)2x+2y=−2

Ch. 5 - Systems of Equations and Inequalities

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

문제 39

6장, 문제 39

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\) y = (x + 3)^2 \\ x + 2y = -2 \(\end{cases}$

검증된 단계별 안내

Start with the given system of equations: $y = (x+3)^2$ and $x + 2y = -2$.

Since $y$ is already expressed in terms of $x$ in the first equation, substitute $y = (x+3)^2$ into the second equation to eliminate $y$.

After substitution, the second equation becomes $x + 2(x+3)^2 = -2$. Expand the squared term $(x+3)^2$ to get $x + 2(x^2 + 6x + 9) = -2$.

Distribute the 2 across the terms inside the parentheses: $x + 2x^2 + 12x + 18 = -2$.

Combine like terms and rearrange the equation to standard quadratic form: $2x^2 + 13x + 18 + 2 = 0$, which simplifies to $2x^2 + 13x + 20 = 0$. Then solve this quadratic equation for $x$ using factoring, completing the square, or the quadratic formula.

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영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Systems of Equations

A system of equations consists of two or more equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Solutions can be points where the graphs of the equations intersect.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it easier to solve.

Quadratic Functions and Their Graphs

A quadratic function, like y = (x + 3)^2, graphs as a parabola. Understanding its shape and properties helps in visualizing solutions to systems involving quadratics and linear equations, as their intersections represent the system's solutions.

In Exercises 29–42, solve each system by the method of your choice. {y=(x+3)2x+2y=−2\(\begin{cases}\) y = (x + 3)^2 \\ x + 2y = -2 \(\end{cases}\){y=(x+3)2x+2y=−2