Q. \(\,x^2 – x – 6\,\)

Q: Solve x^2-x-6=0 using factoring.

Set (x-3)(x+2)=0. Then x=3 or x=-2.

Q: Find the roots of x^2-x-6 using the quadratic formula.

x=\frac{1\pm\sqrt{1+24}}{2}=\frac{1\pm 5}{2}, so x=3 or x=-2.

Q: Complete the square for x^2-x-6.

x^2-x-6=\left(x-\frac{1}{2}\right)^2-\frac{25}{4}.

Q: What are the x-intercepts of y=x^2-x-6?

Intercepts where y=0: (3,0) and (-2,0).

Q: Compute the discriminant of x^2-x-6.

For x^2+bx+c, \Delta=b^2-4ac=(-1)^2-4(1)(-6)=25.

Answer

\(x^2-x-6\) factors as \((x-3)(x+2)\).

Set each factor to zero:

\(x-3=0 \Rightarrow x=3\), and \(x+2=0 \Rightarrow x=-2\).

Final result: \(x=3\) or \(x=-2\).

Detailed Explanation

We want to solve the expression \(x^2 – x – 6\). A common goal for a quadratic expression like this is to factor it (and we can then find its zeros if needed).

Step 1: Identify the quadratic form

The expression is

\[x^2 – x – 6\]

This matches the general quadratic form \(ax^2 + bx + c\) where:

\(a = 1\), \(b = -1\), and \(c = -6\).

Step 2: Factor the quadratic

We look for two numbers that:

Multiply to \(ac = 1 \cdot (-6) = -6\).
Add to \(b = -1\).

The pair of integers that works is \(-3\) and \(2\), because:

\((-3)(2) = -6\)

\((-3) + 2 = -1\)

Step 3: Rewrite using those two numbers

Split the middle term \(-x\) using \(-3\) and \(2\):

\[x^2 – x – 6 = x^2 – 3x + 2x – 6\]

Step 4: Factor by grouping

Group the terms:

\[x^2 – 3x + 2x – 6 = (x^2 – 3x) + (2x – 6)\]

Now factor each group:

\[x^2 – 3x = x(x – 3)\]

\[2x – 6 = 2(x – 3)\]

Step 5: Combine the common factor

Both groups contain \((x – 3)\), so factor it out:

\[x(x – 3) + 2(x – 3) = (x + 2)(x – 3)\]

Final result (factored form)

\[x^2 – x – 6 = (x + 2)(x – 3)\]

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Algebra FAQ

Factor \(x^2-x-6\) into linear factors.

\(x^2-x-6=(x-3)(x+2)\).

Solve \(x^2-x-6=0\) using factoring.

Set \((x-3)(x+2)=0\). Then \(x=3\) or \(x=-2\).

Find the roots of \(x^2-x-6\) using the quadratic formula.

\(x=\frac{1\pm\sqrt{1+24}}{2}=\frac{1\pm 5}{2}\), so \(x=3\) or \(x=-2\).

Complete the square for \(x^2-x-6\).

\(x^2-x-6=\left(x-\frac{1}{2}\right)^2-\frac{25}{4}\).

What are the \(x\)-intercepts of \(y=x^2-x-6\)?

Intercepts where \(y=0\): \((3,0)\) and \((-2,0)\).

Compute the discriminant of \(x^2-x-6\).

For \(x^2+bx+c\), \(\Delta=b^2-4ac=(-1)^2-4(1)(-6)=25\).

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