Follow The Steps To Solve The Equation 2 X − 7 − 3 X − 12 = 0 \sqrt{2x-7} - \sqrt{3x-12} = 0 2 X − 7 ​ − 3 X − 12 ​ = 0 .First, Isolate A Radical And Then Square Both Sides. Choose The Resulting Equation.A. 2 X − 7 − 3 X − 12 = 0 2x - 7 - 3x - 12 = 0 2 X − 7 − 3 X − 12 = 0 B. 2 X − 7 = − 3 X − 12 2x - 7 = -3x - 12 2 X − 7 = − 3 X − 12 C. $2x - 7 = 3x -

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**Solving Equations with Radicals: A Step-by-Step Guide** ===========================================================

Introduction

Solving equations with radicals can be a challenging task, but with the right approach, it can be made easier. In this article, we will focus on solving the equation 2x−7−3x−12=0\sqrt{2x-7} - \sqrt{3x-12} = 0. We will follow the steps to isolate a radical and then square both sides to solve for x.

Step 1: Isolate a Radical

To solve the equation, we need to isolate one of the radicals. We can do this by moving the other radical to the other side of the equation.

Isolate the First Radical

2x−7−3x−12=0\sqrt{2x-7} - \sqrt{3x-12} = 0

Add 3x−12\sqrt{3x-12} to both sides:

2x−7=3x−12\sqrt{2x-7} = \sqrt{3x-12}

Isolate the Second Radical

Now, we can square both sides of the equation to eliminate the radicals.

Step 2: Square Both Sides

Squaring both sides of the equation will eliminate the radicals and give us a new equation to solve.

Square Both Sides

(2x−7)2=(3x−12)2(\sqrt{2x-7})^2 = (\sqrt{3x-12})^2

2x−7=3x−122x-7 = 3x-12

Simplify the Equation

Now, we can simplify the equation by combining like terms.

2x−7=3x−122x-7 = 3x-12

Subtract 2x from both sides:

−7=x−12-7 = x-12

Add 12 to both sides:

5=x5 = x

Check the Solution

Now that we have found the solution, we need to check it to make sure it is correct.

Q&A

Q: Why do we need to isolate a radical? A: We need to isolate a radical because squaring both sides of the equation will eliminate the radicals and give us a new equation to solve.

Q: What happens if we don't isolate a radical? A: If we don't isolate a radical, we will end up with a more complicated equation that is harder to solve.

Q: Can we square both sides of the equation without isolating a radical? A: No, we cannot square both sides of the equation without isolating a radical. This will lead to a more complicated equation that is harder to solve.

Q: How do we know if our solution is correct? A: We know if our solution is correct by checking it in the original equation. If the solution satisfies the original equation, then it is correct.

Conclusion

Solving equations with radicals can be a challenging task, but with the right approach, it can be made easier. By following the steps to isolate a radical and then square both sides, we can solve equations with radicals. Remember to check your solution to make sure it is correct.

Common Mistakes

  • Not isolating a radical before squaring both sides
  • Squaring both sides without isolating a radical
  • Not checking the solution to make sure it is correct

Tips and Tricks

  • Make sure to isolate a radical before squaring both sides
  • Check your solution to make sure it is correct
  • Use a calculator to check your solution if necessary

Practice Problems

  • Solve the equation x+3−x−2=0\sqrt{x+3} - \sqrt{x-2} = 0
  • Solve the equation 2x−5+x+3=0\sqrt{2x-5} + \sqrt{x+3} = 0
  • Solve the equation x−1−x+2=0\sqrt{x-1} - \sqrt{x+2} = 0

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer