Q. \(\;x^2-11x+24\;\)

Q: Find the roots of x^2-11x+24=0.

From (x-3)(x-8)=0, roots are x=3 and x=8.

Q: Solve x^2-11x+24=0 using the quadratic formula.

x=\frac{11\pm\sqrt{121-96}}{2}=\frac{11\pm 5}{2}, so x=3,8.

Q: What is the discriminant of x^2-11x+24?

b^2-4ac= (-11)^2-4(1)(24)=121-96=25.

Q: What are the vertex and minimum value of x^2-11x+24?

Vertex at x=\frac{11}{2}. Minimum value: -\frac{25}{4}.

Answer

We factor the quadratic \(x^2-11x+24\). Find two numbers that multiply to \(24\) and add to \(-11\): \(-3\) and \(-8\).

\[
x^2-11x+24=(x-3)(x-8)
\]

So the roots are \(x=3\) and \(x=8\).

Final: \(x^2-11x+24=(x-3)(x-8)\), with solutions \(x=3,\,8\).

Detailed Explanation

Step 1: Identify the type of expression

The expression \(x^2 – 11x + 24\) is a quadratic polynomial. A common goal is to factor it into a product of two linear factors.

Step 2: Determine numbers that multiply to the constant term

For a quadratic of the form \(ax^2 + bx + c\), here \(a = 1\), \(b = -11\), and \(c = 24\).

We look for two integers whose product is \(24\) and whose sum is \(-11\).

The factor pairs of \(24\) are:

\(1 \cdot 24 = 24\)

\(2 \cdot 12 = 24\)

\(3 \cdot 8 = 24\)

\(4 \cdot 6 = 24\)

To get a sum of \(-11\), we use negative numbers:

\(-3 + (-8) = -11\)

So the two numbers are \(-3\) and \(-8\).

Step 3: Write the factored form

If the two numbers are \(-3\) and \(-8\), then the quadratic factors as

\[
x^2 – 11x + 24 = (x – 3)(x – 8).
\]

Final Answer

\[
x^2 – 11x + 24 = (x – 3)(x – 8).
\]

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Algebra FAQ

Factor \(x^2-11x+24\).

\(x^2-11x+24=(x-3)(x-8)\).

Find the roots of \(x^2-11x+24=0\).

From \( (x-3)(x-8)=0\), roots are \(x=3\) and \(x=8\).

Solve \(x^2-11x+24=0\) using the quadratic formula.

\(x=\frac{11\pm\sqrt{121-96}}{2}=\frac{11\pm 5}{2}\), so \(x=3,8\).

What is the discriminant of \(x^2-11x+24\)?

\(b^2-4ac= (-11)^2-4(1)(24)=121-96=25\).

Complete the square for \(x^2-11x+24\).

\(x^2-11x+24=\left(x-\frac{11}{2}\right)^2-\frac{25}{4}=\left(x-\frac{11}{2}\right)^2-\left(\frac{5}{2}\right)^2\).

What are the vertex and minimum value of \(x^2-11x+24\)?

Vertex at \(x=\frac{11}{2}\). Minimum value: \(-\frac{25}{4}\).

Use math tools to solve x^2-11x+24.
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