Q. \(x^2 – 2x – 24\)

Q: What are the factors of x^2-2x-24?

Find two numbers with product -24 and sum -2: -6 and 4. So x^2-2x-24=(x-6)(x+4).

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

Using (x-6)(x+4)=0, set each factor to zero: x-6=0 gives x=6, and x+4=0 gives x=-4.

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

For a=1,b=-2,c=-24: x=\frac{2\pm\sqrt{(-2)^2-4(1)(-24)}}{2}=\frac{2\pm\sqrt{100}}{2}=1\pm 5. So x=6,-4.

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

D=b^2-4ac=(-2)^2-4(1)(-24)=4+96=100. Since D>0, there are two real distinct roots.

Q: What is the vertex of y=x^2-2x-24?

Vertex x-coordinate is h=-\frac{b}{2a}=-\frac{-2}{2}=1. Then y=1^2-2(1)-24=-25. So vertex is (1,-25).

Q: What is the y-intercept of x^2-2x-24?

Set x=0: y=0-0-24=-24. So the y-intercept is (0,-24).

Q: What is the leading behavior of x^2-2x-24?

Since the leading term is x^2 with positive coefficient, the parabola opens upward. So y\to\infty as x\to\pm\infty.

Answer

We factor the quadratic \(x^2-2x-24\). We need two numbers that multiply to \(-24\) and add to \(-2\). Those numbers are \(-6\) and \(4\).

\[
x^2-2x-24=(x-6)(x+4)
\]

So the factored form is \((x-6)(x+4)\).

Detailed Explanation

We want to work with the quadratic expression \(x^2 – 2x – 24\). A common goal is to factor it, if possible.

Step 1: Write the expression to be factored

We start with:

\[
x^2 – 2x – 24
\]

Step 2: Use the factor idea

For a quadratic of the form \(x^2 + bx + c\), we look for two numbers that multiply to \(c\) and add to \(b\). Here:

\[
x^2 – 2x – 24
\]

So, in this case:

\[
b = -2,\quad c = -24
\]

Step 3: Find numbers that multiply to \(-24\) and add to \(-2\)

We need two integers \(m\) and \(n\) such that:

\[
m n = -24
\]

and

\[
m + n = -2
\]

List factor pairs of \(-24\):

\[
(-1, 24),\ (1, -24),\ (2, -12),\ (-2, 12),\ (3, -8),\ (-3, 8),\ (4, -6),\ (-4, 6)
\]

Now check which pair adds to \(-2\):

\[
2 + (-12) = -10
\]

\[
4 + (-6) = -2
\]

So the correct pair is \(4\) and \(-6\).

Step 4: Build the factored form

Since we found \(4\) and \(-6\), the factorization is:

\[
x^2 – 2x – 24 = (x + 4)(x – 6)
\]

Final Answer

\[
x^2 – 2x – 24 = (x + 4)(x – 6)
\]

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Algebra FAQ

What are the factors of \(x^2-2x-24\)?

Find two numbers with product \(-24\) and sum \(-2\): \(-6\) and \(4\). So \(x^2-2x-24=(x-6)(x+4)\).

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

Using \((x-6)(x+4)=0\), set each factor to zero: \(x-6=0\) gives \(x=6\), and \(x+4=0\) gives \(x=-4\).

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

For \(a=1,b=-2,c=-24\): \(x=\frac{2\pm\sqrt{(-2)^2-4(1)(-24)}}{2}=\frac{2\pm\sqrt{100}}{2}=1\pm 5\). So \(x=6,-4\).

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

\(D=b^2-4ac=(-2)^2-4(1)(-24)=4+96=100\). Since \(D>0\), there are two real distinct roots.

What is the vertex of \(y=x^2-2x-24\)?

Vertex \(x\)-coordinate is \(h=-\frac{b}{2a}=-\frac{-2}{2}=1\). Then \(y=1^2-2(1)-24=-25\). So vertex is \((1,-25)\).

What is the \(y\)-intercept of \(x^2-2x-24\)?

Set \(x=0\): \(y=0-0-24=-24\). So the \(y\)-intercept is \((0,-24)\).

What is the leading behavior of \(x^2-2x-24\)?

Since the leading term is \(x^2\) with positive coefficient, the parabola opens upward. So \(y\to\infty\) as \(x\to\pm\infty\).

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