Q. \(1-x^2\)

Q: Solve the inequality 1-x^2\ge 0.

1-x^2\ge 0 \Rightarrow x^2\le 1 \Rightarrow -1\le x\le 1

Answer

To factor the expression, use the difference of squares formula:

\[
1-x^2 = (1)^2 – (x)^2 = (1-x)(1+x)
\]

Final result:

\[
1-x^2 = (1-x)(1+x)
\]

Detailed Explanation

We want to simplify the expression \(1-x^2\).

Step 1: Recognize the structure as a difference of squares.

An expression of the form \(a^2-b^2\) can be factored as \((a-b)(a+b)\).

Here, \(1-x^2\) can be rewritten as \(1^2-(x)^2\).

Step 2: Identify \(a\) and \(b\).

\(a=1\) and \(b=x\).

Step 3: Apply the difference of squares formula.

So,

\[
1-x^2 = 1^2 – x^2 = (1-x)(1+x).
\]

Final Answer:

\[
1-x^2 = (1-x)(1+x).
\]

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Algebra FAQ

Factor the expression \(1-x^2\).

\(1-x^2=(1-x)(1+x)\)

Simplify \(1-x^2\) in terms of difference of squares.

\(1-x^2=a^2-b^2=(1)^2-(x)^2=(1-x)(1+x)\)

Solve \(1-x^2=0\).

\(1-x^2=0 \Rightarrow x^2=1 \Rightarrow x=\pm 1\)

Find the derivative of \(1-x^2\).

\(\frac{d}{dx}(1-x^2)=0-2x=-2x\)

Evaluate \(1-x^2\) at \(x=3\).

\(1-3^2=1-9=-8\)

Solve the inequality \(1-x^2>0\).

\(1-x^2>0 \Rightarrow x^2<1 \Rightarrow -1

Solve the inequality \(1-x^2\ge 0\).

\(1-x^2\ge 0 \Rightarrow x^2\le 1 \Rightarrow -1\le x\le 1\)

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