Checking a math solution step by step

• State the original claim and all domain constraints before manipulating an equation.
• Classify each transformation as reversible or potentially solution-expanding.
• Verify every candidate in the original statement and report rejected candidates explicitly.

Build the idea

Worked example

1. Step 1

   The radical is defined for x greater than or equal to negative five. Because the principal square root is nonnegative, equality also requires x minus one to be nonnegative, so any solution must satisfy the stronger condition x greater than or equal to one.

   $$x+5\geq0,\quad x-1\geq0\quad\Longrightarrow\quad x\geq1$$

2. Step 2

   Square both sides to remove the radical. This implication is useful but not fully reversible for arbitrary signs, so roots of the quadratic remain candidates.

   $$x+5=(x-1)^2=x^2-2x+1$$

3. Step 3

   Move every term to one side and factor. The zero-product property gives two algebraic candidates, four and negative one.

   $$x^2-3x-4=(x-4)(x+1)=0$$

4. Step 4

   Substitute four into the original. The left side is the square root of nine, or three, and the right side is also three, so four passes.

   $$\sqrt{4+5}=3=4-1$$

5. Step 5

   Negative one violates the recorded constraint and fails direct substitution: the left side is two while the right side is negative two.

   $$\sqrt{-1+5}=2\ne -2=-1-1$$

• Checking candidates only in the squared equation because that is the last equation solved.

  Squaring can expand the solution set. Substitute into the untouched original statement, where sign and domain information are still present, before calling a candidate a solution.

• Erasing a rejected candidate without recording where it came from or why it failed.

  Keep an audit trail: label the value as an algebraic candidate, show the failed constraint or substitution, and state that it is extraneous. This makes the reasoning reviewable.

Solve and check the square root of the quantity x plus six, end quantity, equals x, explicitly testing every candidate from the squared equation.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Preserve the original problem

   Start by copying the square-root equation exactly. Because a principal square root cannot be negative, x minus one must be nonnegative, so x must be at least one. Write that condition beside the original. It will remain available even after later algebra hides the square-root sign information.

   • The square root of the quantity x plus five, end quantity, equals x minus one.
   • The radical domain requires x to be greater than or equal to negative five.
   • x plus five must be nonnegative, and x minus one must be nonnegative, so a solution must have x greater than or equal to one.
2. Part 2

   Mark squaring as a candidate generator

   Square both sides and simplify to a quadratic. Squaring preserves equality for a true original statement, but equal squares do not guarantee equal signed quantities. Therefore this step may add values. We can solve the quadratic, but its roots must be labeled candidates until the original equation accepts them.

   • x plus five equals the quantity x minus one, squared, which equals x squared minus two x plus one.
   • x squared minus three x minus four equals the quantity x minus four times the quantity x plus one, which equals zero.
3. Part 3

   Return to the untouched equation

   Substitute four into the original: both sides equal three, so four passes. Substitute negative one: the square root side is positive two while the right side is negative two. Negative one also violates the earlier sign constraint. It is an explainable extraneous candidate, not a solution to erase silently.

   • The square root of the quantity four plus five, end quantity, equals three, which equals four minus one.
   • The square root of the quantity negative one plus five, end quantity, equals two, which is not equal to negative two, the value of negative one minus one.
4. Part 4

   State the verified conclusion

   Report that the candidate set from the quadratic is four and negative one, while the verified solution set contains only four. Name squaring as the step that allowed the extra candidate and show the failed substitution. This short audit trail lets another learner review the claim, transformation, and evidence without trusting the final line on appearance.

   • The square root of the quantity x plus five, end quantity, equals x minus one.
   • x squared minus three x minus four equals the quantity x minus four times the quantity x plus one, which equals zero.

Scope and limitations

• This lesson demonstrates equation checking and does not replace a formal proof when a problem asks for a general theorem or all cases of an argument.
• Numerical substitution can confirm a proposed value but cannot establish that no other solutions exist unless the preceding solution method is complete.