Math Problem Statement

Solve the given radical and quadratic equations as shown in the uploaded document.

Solution

Thank you for sharing the image. I will extract and solve the math problems step by step. Let’s start with the solutions for Nos. 1–6, which involve solving radical equations.

1. t5=2\sqrt{t} - 5 = 2

  • Isolate the square root term:
    t=7\sqrt{t} = 7
  • Square both sides:
    t=49t = 49

2. y=5/yy = 5/\sqrt{y}

  • Multiply both sides by y\sqrt{y} (assuming y>0y > 0):
    yy=5y\sqrt{y} = 5
  • Rewrite as y=5/y\sqrt{y} = 5/y. Square both sides: y=25y = 25

3. 3x6x1=33x - 6\sqrt{x-1} = 3

  • Isolate the square root term:
    6x1=3x36\sqrt{x-1} = 3x - 3
  • Divide through by 3:
    2x1=x12\sqrt{x-1} = x - 1
  • Square both sides:
    4(x1)=(x1)24(x-1) = (x-1)^2
  • Expand and simplify:
    4x4=x22x+14x - 4 = x^2 - 2x + 1
    x26x+5=0x^2 - 6x + 5 = 0
  • Solve x=1x = 1 and x=5x = 5. Check for extraneous solutions.

Would you like me to continue with the remaining questions or clarify anything so far?

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Math Problem Analysis

Mathematical Concepts

Radical Equations
Quadratic Equations
Factoring
Substitution Methods

Formulas

Isolating radicals and squaring both sides
Factoring quadratic equations
Substitution to simplify equations

Theorems

Properties of Equality
Zero-Product Property

Suitable Grade Level

Grades 10-12