Problem 44 Solve each equation and check fo... [FREE SOLUTION]

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Chapter 7: Problem 44

Solve each equation and check for extraneous solutions. $$\sqrt{2 x^{2}+6 x+9}=x$$

Short Answer

Expert verified

There are no valid solutions since -3 is extraneous.

Step by step solution

Set up the equation

Start with the given equation: $\sqrt{2x^2 + 6x + 9} = x$.

Square both sides

To eliminate the square root, square both sides of the equation: $(\sqrt{2x^2 + 6x + 9})^2 = x^2$. This simplifies to $(2x^2 + 6x + 9) = x^2$.

Simplify the equation

Subtract $x^2$ from both sides to form a standard quadratic equation: $(2x^2 + 6x + 9 - x^2) = 0$ which simplifies to $x^2 + 6x + 9 = 0$.

Factor the quadratic equation

Factor the quadratic equation: $(x + 3)^2 = 0$.

Solve for x

Set the factor equal to zero and solve for $x$: $(x + 3) = 0 $ which gives $x = -3$.

Check for extraneous solutions

To check for extraneous solutions, substitute $x = -3$ back into the original equation: Left side: $\sqrt{2(-3)^2 + 6(-3) + 9} = \sqrt{18 - 18 + 9} = \sqrt{9} = 3$ Right side: $(-3)$ Since $\sqrt{9} eq -3$, $x = -3$ is an extraneous solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Equations

A square root equation is an equation where a variable appears inside a square root. The general form looks like $\text{expression with x inside square root} = \text{another expression}$. Solving these equations usually involves isolating the square root term and then squaring both sides to eliminate the square root.

Extraneous Solutions

Extraneous solutions are solutions that arise from the algebraic process of solving the equation but are not valid when substituted back into the original equation. In our example, we identified that substituting $\text{x = -3} $ into the original equation does not hold true. Hence, it鈥檚 important always to check potential solutions in the original equation to identify any extraneous results.

Factoring Quadratic Equations

Factoring quadratic equations is a method to solve equations of the form $\text{ax}^2 + \text{bx} + \text{c} = 0$. The equation can often be written as $ (\text{px + q})(\text{rx + s}) = 0$. In our example, we ended up with $ (x + 3)^2 = 0$ and solved it by setting $x + 3 = 0 $ which gave $ x = -3 $.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate the variable of interest. This includes operations such as squaring both sides of an equation, adding or subtracting terms, and factoring. In solving our original equation $ \sqrt{2 x^{2}+6 x+9}=x$, we squared both sides and then simplified to form a quadratic equation that could be factored and solved.

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91影视

Solve each equation and check for extraneous solutions. $$\sqrt{2 x^{2}+6 x+9}=x$$

Short Answer

Step by step solution

Set up the equation

Square both sides

Simplify the equation

Factor the quadratic equation

Solve for x

Check for extraneous solutions

Key Concepts

Square Root Equations

Extraneous Solutions

Factoring Quadratic Equations

Algebraic Manipulation

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