Upward parabola; positive outside roots. x 2.

Q. \(\,x^2-3x+2\,\)

Q: Find the zeros of x^2-3x+2=0.

Solve (x-1)(x-2)=0\Rightarrow x=1 or x=2.

Q: Compute the vertex and minimum value of x^2-3x+2.

Vertex at x=\frac{-(-3)}{2\cdot1}=\frac{3}{2}. Minimum value f\left(\frac{3}{2}\right)=\frac{1}{4}.

Q: Determine the discriminant of x^2-3x+2.

a=1,b=-3,c=2. \Delta=b^2-4ac=9-8=1, so two real distinct roots.

Answer

We factor the quadratic.

\[
x^2-3x+2=(x-1)(x-2)
\]

So the solutions come from each factor being zero.

\[
x-1=0 \Rightarrow x=1,\quad x-2=0 \Rightarrow x=2
\]

Final answer: \(\,x=1\) or \(x=2\).

Detailed Explanation

We are asked to simplify or factor the expression \(x^2 – 3x + 2\). A common goal for a quadratic like this is to factor it into the form \((x-a)(x-b)\).

Step 1: Identify the coefficients.

The expression is:

\[x^2 – 3x + 2\]

So the coefficients are:

\[a = 1,\quad b = -3,\quad c = 2\]

Step 2: Find two numbers that multiply to \(c\) and add to \(b\).

We need numbers \(m\) and \(n\) such that:

\[m \cdot n = 2\]

\[m + n = -3\]

Check factor pairs of \(2\): \(1 \cdot 2 = 2\) and \((-1) \cdot (-2) = 2\).

Now see which pair adds to \(-3\): the numbers \(-1\) and \(-2\) add to \(-3\).

So we choose:

\[m = -1,\quad n = -2\]

Step 3: Write the factored form.

Using \(m\) and \(n\), the quadratic factors as:

\[x^2 – 3x + 2 = (x + m)(x + n)\]

Substitute \(m = -1\) and \(n = -2\):

\[x^2 – 3x + 2 = (x – 1)(x – 2)\]

Final Answer:

\[x^2 – 3x + 2 = (x – 1)(x – 2)\]

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Algebra FAQ

Factor \(x^2-3x+2\).

\(x^2-3x+2=(x-1)(x-2)\).

Find the zeros of \(x^2-3x+2=0\).

Solve \((x-1)(x-2)=0\Rightarrow x=1\) or \(x=2\).

Solve \(x^2-3x+2>0\).

Upward parabola; positive outside roots. \(x<1\) or \(x>2\).

Solve \(x^2-3x+2<0\).

Negative between roots. \(1

Compute the vertex and minimum value of \(x^2-3x+2\).

Vertex at \(x=\frac{-(-3)}{2\cdot1}=\frac{3}{2}\). Minimum value \(f\left(\frac{3}{2}\right)=\frac{1}{4}\).

Determine the discriminant of \(x^2-3x+2\).

\(a=1,b=-3,c=2\). \(\Delta=b^2-4ac=9-8=1\), so two real distinct roots.

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