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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 18
Chapter 6, Problem 18

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
y=x2+6x+9y=x^2+6x+9
x+2y=−2x+2y=-2

Verified step by step guidance
1
Start with the given system of equations: \(y = x^{2} + 6x + 9\) and \(x + 2y = -2\).
Substitute the expression for \(y\) from the first equation into the second equation to eliminate \(y\). This gives: \(x + 2(x^{2} + 6x + 9) = -2\).
Expand and simplify the equation: \(x + 2x^{2} + 12x + 18 = -2\). Combine like terms to get a quadratic equation in terms of \(x\).
Bring all terms to one side to set the quadratic equation equal to zero: \(2x^{2} + 13x + 20 = 0\).
Solve the quadratic equation using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a = 2\), \(b = 13\), and \(c = 20\). After finding the values of \(x\), substitute each back into the original equation for \(y\) to find the corresponding \(y\) values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Nonlinear Systems of Equations

A nonlinear system involves at least one equation that is not linear, such as quadratic or higher-degree polynomials. Solving these systems requires methods beyond simple substitution or elimination used for linear systems, often involving substitution of one equation into another to reduce the system to a single-variable equation.
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Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved using algebraic techniques, including factoring or the quadratic formula.
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Complex Solutions

When solving polynomial equations, solutions may include complex numbers if the discriminant is negative. Complex solutions have a real part and an imaginary part, and it is important to include these when the problem specifies all solutions, not just real ones.
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Related Practice
Textbook Question

Write the system of equations associated with each augmented matrix . Do not solve.

[10020103001−2]\(\left\)[ \(\begin{matrix}\) 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -2 \(\end{matrix}\) \(\right\)]

1
views
Textbook Question

Graph each inequality. x ≤ 3

5
views
Textbook Question

Find the inverse, if it exists, for each matrix. [−1−234]\(\left\)[ \(\begin{matrix}\) -1 & -2 \\3 & 4 \(\end{matrix}\) \(\right\)]

Textbook Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.

y = x2 - 2x + 1

x - 3y = -1

Textbook Question

Graph each inequality. 2x > 3 - 4y

3
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Textbook Question

Find the inverse, if it exists, for each matrix. [−12−2−1]\(\left\)[ \(\begin{matrix}\) -1 & 2 \\-2 & -1 \(\end{matrix}\) \(\right\)]