Q. \(x^2 – x – 12\)

Q: . Factor x^2-x-12?

. \left(x-4\right)\left(x+3\right)

Q: . Find the zeros of x^2-x-12?

. Solve \left(x-4\right)\left(x+3\right)=0. Zeros are x=4 and x=-3

Q: . Solve x^2-x-12=0 using the quadratic formula?

. x=\dfrac{1\pm\sqrt{1+48}}{2}=\dfrac{1\pm 7}{2}. So x=4 or x=-3

Q: . What is the vertex (complete the square) of x^2-x-12?

. x^2-x-12=\left(x-\tfrac12\right)^2-\tfrac{49}{4}. Vertex is \left(\tfrac12,-\tfrac{49}{4}\right)

Q: . Graphing: where does y=x^2-x-12 cross the x-axis?

. Crossings occur at the roots x=4 and x=-3. Points: (4,0) and (-3,0)

Q: . Compute the y-intercept of x^2-x-12?

. Set x=0: y=-12. So the y-intercept is (0,-12)

Answer

We factor the quadratic \(x^2 – x – 12\). Find two numbers that multiply to \(-12\) and add to \(-1\): \(-4\) and \(3\).

\[
x^2 – x – 12 = (x – 4)(x + 3)
\]

So the solutions come from \((x – 4)=0\) or \((x + 3)=0\):

\[
x = 4 \quad \text{or} \quad x = -3
\]

Final result: \(x^2 – x – 12 = (x – 4)(x + 3)\), with \(x = 4\) or \(x = -3\).

Detailed Explanation

We want to work with the expression \(x^2 – x – 12\). A common goal is to factor it into simpler parts. That is usually done by finding two numbers that multiply to \(-12\) and add to \(-1\).

Step 1: Identify the form.

The expression \(x^2 – x – 12\) matches the quadratic form \(x^2 + bx + c\), where \(b = -1\) and \(c = -12\).

Step 2: Find two numbers.

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

\(m \cdot n = -12\)
\(m + n = -1\)

Step 3: List factor pairs of \(-12\).

Possible pairs that multiply to \(-12\) are:

\(3\) and \(-4\), since \(3 \cdot (-4) = -12\)
\(-3\) and \(4\), since \((-3) \cdot 4 = -12\)

Check their sums:

\(3 + (-4) = -1\)
\((-3) + 4 = 1\)

The pair that gives sum \(-1\) is \(3\) and \(-4\).

Step 4: Factor the quadratic.

So we rewrite \(x^2 – x – 12\) using these numbers:

\[
x^2 – x – 12 = (x + 3)(x – 4)
\]

Final Answer:

\[
x^2 – x – 12 = (x + 3)(x – 4)
\]

See full solution

Graph

Need help with x²−x−12? Try our AI homework tools!

Homework Helper

Algebra FAQ

. Factor \(x^2-x-12\)?

. \(\left(x-4\right)\left(x+3\right)\)

. Find the zeros of \(x^2-x-12\)?

. Solve \(\left(x-4\right)\left(x+3\right)=0\). Zeros are \(x=4\) and \(x=-3\)

. Solve \(x^2-x-12=0\) using the quadratic formula?

. \(x=\dfrac{1\pm\sqrt{1+48}}{2}=\dfrac{1\pm 7}{2}\). So \(x=4\) or \(x=-3\)

. What is the vertex (complete the square) of \(x^2-x-12\)?

. \(x^2-x-12=\left(x-\tfrac12\right)^2-\tfrac{49}{4}\). Vertex is \(\left(\tfrac12,-\tfrac{49}{4}\right)\)

. Graphing: where does \(y=x^2-x-12\) cross the \(x\)-axis?

. Crossings occur at the roots \(x=4\) and \(x=-3\). Points: \((4,0)\) and \((-3,0)\)

. Compute the \(y\)-intercept of \(x^2-x-12\)?

. Set \(x=0\): \(y=-12\). So the \(y\)-intercept is \((0,-12)\)

Use an AI tool to check steps.
Solve x^2-x-12 quickly.

298,376+ active customers
Math, Geometry, Trigonometry, etc.

AI Math Solver Geometry AI Trig Solver