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Chapter 10: Problem 12

Solve each equation. $$ \sqrt{5 k-3}+2=0 $$

Short Answer

Expert verified
No real solution.

Step by step solution

01

Isolate the Square Root

Subtract 2 from both sides of the equation to isolate the square root term: \[\sqrt{5k - 3} + 2 = 0\]\[\sqrt{5k - 3} = -2\]
02

Evaluate the Isolated Square Root

Since the square root of a number cannot be negative, \(\sqrt{5k - 3} = -2\) has no real solution. This indicates that no real number \(k\) satisfies the original equation.

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Key Concepts

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

Isolating Square Root
In algebra, isolating the square root is a crucial step. This means getting the square root term by itself on one side of the equation. For example, in the equation \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sqrt{5k - 3} + 2 = 0\ \ ,\ \ you need to remove the \ \ \ \ + 2\ \ from the equation. \ \ \ \ \ Subtract 2 from both sides: \ \ \[ \sqrt{5k - 3} + 2 - 2 = 0 - 2 \] \ Now it's isolated: \[ \sqrt{5k - 3} = -2 \] \ This step is essential because it makes the equation simpler to handle.
No Real Solution
If you isolate the square root and end up with a negative number like \[ \sqrt{5k - 3} = -2 \] \ , you have a problem. The square root of a number is never negative. In mathematics, \ \ \sqrt{x}\ \ \ is defined as the non-negative number that, when multiplied by itself, gives \x\. If you ever isolate a square root and see a negative result, immediately recognize the equation has no real solution. For this example, \[ \sqrt{5k - 3} = -2 \] \ tells us there is no real \k\ that will satisfy the equation, leading to the conclusion:
    \ \ \ \ \ \ \ \ \ \
  • No real solution
  • No real number exists for \ \ \ k \ to make \[ \sqrt{5k - 3} = -2 \]
Algebra Basics
Understanding algebra basics is key to solving square root equations. Here are some core principles to remember:
  • Always perform the same operation on both sides of the equation to maintain equality.
  • To isolate terms, use inverse operations like subtraction if there鈥檚 addition.
  • Remember, \[ \sqrt{x} \] represents only the non-negative root of x.

    Practice isolating terms step-by-step to build a strong foundation in algebra. Concepts like isolating the square root, recognizing non-real solutions, and the importance of maintaining balance in an equation are very fundamental.

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