Question

what method would you choose to solve the equation (2x^2 - 7 = 9)? explain why you chose this method.

Explanation:

⚡ Using what you learned: factoring and solving quadratic equations

Step 1: Analyze the structure of the equation

The given equation is:

$$ 2x^2 - 7 = 9 $$

This is a quadratic equation because the highest power of the variable \( x \) is \( 2 \). Notice that there is no linear term (no \( x \) term with a power of \( 1 \)). The equation only contains an \( x^2 \) term and constant numbers.

Step 2: Determine the most efficient method

When a quadratic equation has no \( x \) term (it is in the form \( ax^2 + c = 0 \) or can be easily rewritten as \( ax^2 = d \)), the most direct and efficient method to solve it is taking the square root (also known as the square root property).

Other methods like factoring, completing the square, or using the quadratic formula are unnecessary and require more steps because we can easily isolate the \( x^2 \) term.

Step 3: Justify the choice with the solution steps

To show why this method is the easiest, we can isolate \( x^2 \) and solve:

1. Add \( 7 \) to both sides to isolate the variable term:

$$ 2x^2 = 16 $$

1. Divide both sides by \( 2 \):

$$ x^2 = 8 $$

1. Take the square root of both sides:

$$ x = \pm\sqrt{8} = \pm2\sqrt{2} $$

This confirms that isolating the \( x^2 \) term and taking the square root is the fastest and most straightforward method.

This method is chosen because the equation does not contain a linear \( x \) term (it only has an \( x^2 \) term and constant numbers). This allows you to easily isolate \( x^2 \) on one side of the equation (resulting in \( x^2 = 8 \)) and solve directly by taking the square root of both sides, making it much faster and simpler than factoring, completing the square, or using the quadratic formula.

Answer:

The best method to solve the equation \( 2x^2 - 7 = 9 \) is taking square roots (or isolating the \( x^2 \) term).