Math Problem Statement

sqrt(11-10x) equals (x-2)

Solution

To solve the equation:

1110x=x2\sqrt{11 - 10x} = x - 2

we need to square both sides to eliminate the square root:

(1110x)2=(x2)2(\sqrt{11 - 10x})^2 = (x - 2)^2

Simplifying, we get:

1110x=(x2)211 - 10x = (x - 2)^2

Expanding the right-hand side:

1110x=x24x+411 - 10x = x^2 - 4x + 4

Now, let's bring everything to one side to form a quadratic equation:

x2+6x7=0x^2 + 6x - 7 = 0

This is a standard quadratic equation, and we can solve it using the quadratic formula:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where a=1a = 1, b=6b = 6, and c=7c = -7:

x=6±6241(7)21x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1}

x=6±36+282x = \frac{-6 \pm \sqrt{36 + 28}}{2}

x=6±642x = \frac{-6 \pm \sqrt{64}}{2}

x=6±82x = \frac{-6 \pm 8}{2}

Now, calculate the two possible values of xx:

  1. x=6+82=22=1x = \frac{-6 + 8}{2} = \frac{2}{2} = 1
  2. x=682=142=7x = \frac{-6 - 8}{2} = \frac{-14}{2} = -7

Verification of Solutions:

We need to verify these solutions because squaring both sides might introduce extraneous solutions:

  1. Check x=1x = 1: 11101=12\sqrt{11 - 10 \cdot 1} = 1 - 2 1=1(False)\sqrt{1} = -1 \quad \text{(False)}

  2. Check x=7x = -7: 1110(7)=72\sqrt{11 - 10 \cdot (-7)} = -7 - 2 81=9(False)\sqrt{81} = -9 \quad \text{(False)}

Conclusion:

Both potential solutions, x=1x = 1 and x=7x = -7, fail upon verification. Therefore, there is no solution to the equation 1110x=x2\sqrt{11 - 10x} = x - 2.

Would you like further details or have any questions about this process?

Here are 5 related questions:

  1. How do we identify extraneous solutions in equations involving square roots?
  2. What happens when we square both sides of an equation?
  3. How can we check the domain restrictions of square root functions?
  4. Why do quadratic equations sometimes produce extraneous solutions?
  5. How can graphing the functions help verify the solution?

Tip: Always verify your solutions when squaring both sides of an equation, as this can introduce solutions that don't satisfy the original equation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Square Root Equations
Quadratic Equations
Extraneous Solutions

Formulas

Square both sides: (sqrt(a))^2 = a
Quadratic equation formula: ax^2 + bx + c = 0
Quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a

Theorems

Quadratic formula
Verification of solutions for square root equations

Suitable Grade Level

Grades 9-12