Q. \(x^2-11x+24=0\)

Q: Solve x^2-11x+24=0 by factoring.

(x-3)(x-8)=0\Rightarrow x=3 \text{ or } x=8.

Q: What is the discriminant b^2-4ac?

Here a=1,b=-11,c=24. Discriminant: (-11)^2-4(1)(24)=121-96=25.

Q: What is the sum and product of roots?

Sum =11 and product =24. For roots 3 and 8: 3+8=11, 3\cdot 8=24.

Q: Check solutions x=3 and x=8.

3^2-11(3)+24=9-33+24=0. 8^2-11(8)+24=64-88+24=0.

Answer

We solve the quadratic \(x^2-11x+24=0\) by factoring:

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

Set each factor equal to \(0\):

\[
x-3=0 \Rightarrow x=3,\quad x-8=0 \Rightarrow x=8
\]

Final answers: \(x=3\) or \(x=8\).

Detailed Explanation

We need to solve the quadratic equation:

\[
x^2 – 11x + 24 = 0
\]

Step 1: Factor the quadratic

For a quadratic of the form

\[
x^2 – 11x + 24
\]

we want to rewrite it as

\[
(x – a)(x – b) = 0
\]

so that

\[
a + b = 11
\]

and

\[
ab = 24
\]

Step 2: Find numbers that multiply to 24 and add to 11

The factor pairs of \(24\) are \(1 \cdot 24\), \(2 \cdot 12\), and \(3 \cdot 8\).

Check which pair adds to \(11\):

\(1 + 24 = 25\) (not 11)
\(2 + 12 = 14\) (not 11)
\(3 + 8 = 11\) (yes!)

Step 3: Write the factored form

Since \(3\) and \(8\) add to \(11\) and multiply to \(24\), we factor as:

\[
x^2 – 11x + 24 = (x – 3)(x – 8)
\]

Step 4: Set each factor equal to zero

The equation becomes:

\[
(x – 3)(x – 8) = 0
\]

For a product to be zero, at least one factor must be zero, so:

\[
x – 3 = 0
\]

\[
x – 8 = 0
\]

Step 5: Solve each simple equation

If \(x – 3 = 0\), then:

\[
x = 3
\]

If \(x – 8 = 0\), then:

\[
x = 8
\]

Final Answer

The solutions to \(x^2 – 11x + 24 = 0\) are:

\[
x = 3 \quad \text{and} \quad x = 8
\]

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Algebra FAQ

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

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

Solve \(x^2-11x+24=0\) by factoring.

\((x-3)(x-8)=0\Rightarrow x=3 \text{ or } x=8\).

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

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

What is the discriminant \(b^2-4ac\)?

Here \(a=1,b=-11,c=24\). Discriminant: \((-11)^2-4(1)(24)=121-96=25\).

What is the sum and product of roots?

Sum \(=11\) and product \(=24\). For roots \(3\) and \(8\): \(3+8=11\), \(3\cdot 8=24\).

Check solutions \(x=3\) and \(x=8\).

\(3^2-11(3)+24=9-33+24=0\). \(8^2-11(8)+24=64-88+24=0\).

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