Answered

Solve the equation by the method of your choice.

$5 x^2=160$



(Simplify your answer. Type an exact answer, using radicals and $i$ as needed. Use integers or fractions for any numbers in the expression. Type each solution only once.)

Verified Answer :

  • Isolate \(x^2\) by dividing the equation \(5x^2 = 160\) by \(5\), resulting in \(x^2 = 32\).
  • Apply the square root property to get \(x = \pm \sqrt{32}\).
  • Simplify the radical \(\sqrt{32}\) as \(4\sqrt{2}\).
  • State the final solution set as \(\boxed{\{-4\sqrt{2}, 4\sqrt{2}\}}\).

Explanation

  1. Isolating the Variable Squared

We are given the equation \(5x^2 = 160\). To find the solution set for \(x\), we need to isolate the variable \(x\). First, we divide both sides of the equation by 5:
\[x^2 = \frac{160}{5}\]
\[x^2 = 32\]

  1. Solving for x and Simplifying the Radical

Now that we have \(x^2 = 32\), we apply the square root property, which states that if \(x^2 = k\), then \(x = \pm\sqrt{k}\). Therefore:
\[x = \pm\sqrt{32}\]
To simplify \(\sqrt{32}\), we identify the largest perfect square factor of 32. Since \(32 = 16 \times 2\) and 16 is a perfect square:
\[\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\]
Thus, the solutions for \(x\) are \(4\sqrt{2}\) and \(-4\sqrt{2}\).

  1. Final Solution Set

The solutions for \(x\) are \(\pm 4\sqrt{2}\). The solution set is expressed as:
\[\boxed{\{-4\sqrt{2}, 4\sqrt{2}\}}\]

Examples

Solving quadratic equations is a fundamental skill used in physics and engineering. For example, if you are calculating the time it takes for an object to fall a certain distance under gravity, you might arrive at an equation like \(d = \frac{1}{2}gt^2\). Knowing how to isolate \(t\) by dividing by the constants and taking the square root allows you to determine the time elapsed for specific heights, helping you analyze motion and energy in real-world scenarios.

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