What is a quadratic equation, and how can it be solved using different methods like factoring, extracting square roots, and completing the square?
Understand the Problem
The question relates to quadratic equations, specifically focusing on solving methods such as factoring, extracting square roots, and completing the square. It provides an example and mentions how to write a quadratic equation in the standard form and solve it using different techniques.
Answer
The solutions to the equation are $x = 6\sqrt{2}$ and $x = -6\sqrt{2}$.
Answer for screen readers
The solutions for the equation $x^2 - 72 = 0$ are: $$ x = 6\sqrt{2} \quad \text{and} \quad x = -6\sqrt{2} $$
Steps to Solve
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Identify the Quadratic Equation The equation given is $x^2 - 72 = 0$.
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Transposing the Constant To isolate the quadratic term, transpose $-72$ to the other side of the equation: $$ x^2 = 72 $$
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Extract the Square Root Now, take the square root of both sides to solve for $x$: $$ x = \pm \sqrt{72} $$
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Simplify the Square Root Since $72$ can be factored into $36 \times 2$, simplify the square root: $$ x = \pm \sqrt{36 \times 2} $$ $$ x = \pm 6\sqrt{2} $$
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Final Solutions The final solutions for $x$ are: $$ x = 6\sqrt{2} \text{ and } x = -6\sqrt{2} $$
The solutions for the equation $x^2 - 72 = 0$ are: $$ x = 6\sqrt{2} \quad \text{and} \quad x = -6\sqrt{2} $$
More Information
The square root of $72$ is simplified because it has a perfect square factor, which helps to express the solution in a simpler form. This method demonstrates how to extract square roots from quadratic equations effectively.
Tips
- Forgetting to transpose when moving terms from one side of the equation to the other. Always remember to change the sign.
- Not simplifying the square root correctly; always check for perfect squares to make the solution easier.
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