Solve each equation by the method of your choice. √2 x² + 3x - 2√2 = 0

Verified step by step guidance

1. Rewrite the given equation: $\backslash sqrt\{2\}x^2\; +\; 3x\; -\; 2\backslash sqrt\{2\}\; =\; 0$. This is a quadratic equation in standard form $ax^2\; +\; bx\; +\; c\; =\; 0$, where $a\; =\; \backslash sqrt\{2\}$, $b\; =\; 3$, and $c\; =\; -2\backslash sqrt\{2\}$.
2. Use the quadratic formula to solve for $x$: $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$. Substitute the values of $a$, $b$, and $c$ into the formula.
3. Simplify the discriminant $b^2\; -\; 4ac$: Compute $b^2$ as $3^2\; =\; 9$, and compute $4ac$ as $4(\backslash sqrt\{2\})(-2\backslash sqrt\{2\})$. Simplify $4ac$ to get $-16$. Then, calculate the discriminant: $b^2\; -\; 4ac\; =\; 9\; -\; (-16)\; =\; 25$.
4. Substitute the discriminant and other values into the quadratic formula: $x\; =\; \backslash frac\{-3\; \backslash pm\; \backslash sqrt\{25\}\}\{2\backslash sqrt\{2\}\}$. Simplify $\backslash sqrt\{25\}$ to $5$, so the equation becomes $x\; =\; \backslash frac\{-3\; \backslash pm\; 5\}\{2\backslash sqrt\{2\}\}$.
5. Split the equation into two cases to find the two possible solutions for $x$: Case 1: $x\; =\; \backslash frac\{-3\; +\; 5\}\{2\backslash sqrt\{2\}\}$, and Case 2: $x\; =\; \backslash frac\{-3\; -\; 5\}\{2\backslash sqrt\{2\}\}$. Simplify each case to find the final solutions.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Square Roots

Square roots are the values that, when multiplied by themselves, yield the original number. In the context of the given equation, the presence of the square root indicates that we may need to isolate the variable or manipulate the equation to eliminate the square root. Mastery of square root properties is crucial for simplifying and solving equations involving them.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied together to obtain the original expression. In solving quadratic equations, factoring can provide a straightforward method to find the roots of the equation. Recognizing patterns and applying techniques such as the difference of squares or grouping are key skills in this process.