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Chapter 8: Problem 26

Use the square root property to solve each equation. \(5 z^{2}-200=0\)

Short Answer

Expert verified
The solutions are \( z = 2\sqrt{10} \) and \( z = -2\sqrt{10} \).

Step by step solution

01

- Isolate the quadratic term

Start with the equation: \[ 5z^2 - 200 = 0 \]Add 200 to both sides to isolate the quadratic term: \[ 5z^2 = 200 \]
02

- Divide by the coefficient of the quadratic term

To solve for \(z^2\), divide both sides of the equation by 5: \[ z^2 = \frac{200}{5} \] This simplifies to: \[ z^2 = 40 \]
03

- Apply the square root property

Take the square root of both sides of the equation to solve for \(z\): \[ z = \pm\sqrt{40} \] Simplify the square root of 40: \[ z = \pm2\sqrt{10} \]

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Key Concepts

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

square root property
The square root property is a useful tool for solving quadratic equations, especially those of the form a{x}^2 + c = 0 . The square root property states that if x^2 = k , then x = 卤 sqrt{ k }. In simpler terms, once you isolate the quadratic term and have an equation like x^2 = k , you can find x by taking the square root of both sides, remembering that you will get two solutions: the positive and the negative square root of k . In our exercise, after simplifying, 5z^2 - 200 = 0 becomes z^2 = 40 . By applying the square root property, we get z = 卤 sqrt{40} , which simplifies to z = 卤 2sqrt{10}.
isolating the quadratic term
To effectively use the square root property, you need to first isolate the quadratic term. This means getting the quadratic term by itself on one side of the equation. Here鈥檚 how you do it:
  • Start with your original equation. In our example, it was 5z^2 - 200 = 0
  • Add or subtract constants to move them to the other side. For our problem, we added 200 to both sides: 5z^2 = 200
  • Divide or multiply to get the quadratic term alone. We divided by 5 to get: z^2 = 40
Isolating the quadratic term allows you to your equation into a form where the square root property can be applied.
simplifying square roots
Once you have applied the square root property, you might get a square root that needs simplification. Simplifying square roots involves breaking them down to their simplest form. Here鈥檚 a closer look:
  • Identify any perfect squares within your square root. For example, the square root of 40 is sqrt{40} . To simplify, recognize that 40 = 4 * 10.
  • Write the square root as the product of two roots: sqrt{40} = sqrt{4} times sqrt{10} .
  • Simplify the perfect square roots. Here, sqrt{4} is 2, so we get 2sqrt{10}
So, sqrt{40} becomes 2 sqrt{10}. Therefore, we use 2sqrt{10} in the final solution: z = 卤 2 sqrt{10}.
algebraic manipulation
Algebraic manipulation plays a key role in solving quadratic equations. It involves using algebraic operations to rewrite equations in a more useful form. Here are the main steps:
  • Add or subtract terms on both sides to balance the equation.
  • Use multiplication or division to isolate your variable or quadratic term. In the original equation, adding 200 and then dividing by 5 helps isolate \( z^2 \).
  • Apply properties like the square root property to solve for the variable. As we isolated $ z^2 , we could finally apply the square root.
It鈥檚 essential to carefully perform each step and check your work as you proceed. Correct algebraic manipulation ensures you keep the equation balanced and reach the correct solution.

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