f(x)=x^2-x+2

g(x)=2x^2-3x-6

If ​f(x) equals ​g(x), then x equals

We are given the two functions:

- \( f(x) = x^2 - x + 2 \)
- \( g(x) = 2x^2 - 3x - 6 \)

We are asked to find the value of \( x \) where \( f(x) = g(x) \).

### Step 1: Set the two functions equal to each other:
\[
x^2 - x + 2 = 2x^2 - 3x - 6
\]

### Step 2: Move all terms to one side to form a quadratic equation:
\[
x^2 - x + 2 - 2x^2 + 3x + 6 = 0
\]
Simplify the equation:
\[
- x^2 + 2x + 8 = 0
\]
Multiply through by \(-1\) to make the \( x^2 \)-term positive:
\[
x^2 - 2x - 8 = 0
\]

### Step 3: Solve the quadratic equation using the quadratic formula.
The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For \( x^2 - 2x - 8 = 0 \), we have:
- \( a = 1 \)
- \( b = -2 \)
- \( c = -8 \)

Now substitute these values into the quadratic formula:
\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-8)}}{2(1)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 32}}{2}
\]
\[
x = \frac{2 \pm \sqrt{36}}{2}
\]
\[
x = \frac{2 \pm 6}{2}
\]

### Step 4: Find the two solutions:
\[
x = \frac{2 + 6}{2} = \frac{8}{2} = 4
\]
\[
x = \frac{2 - 6}{2} = \frac{-4}{2} = -2
\]

### Final answer:
The values of \( x \) that satisfy \( f(x) = g(x) \) are:
\[
x = 4 \quad \text{or} \quad x = -2
\]

Would you like further details on any of these steps or have any questions?

---

Here are 5 related questions to explore:
1. How would the solution change if we altered the coefficients in \( f(x) \)?
2. Can you graph \( f(x) \) and \( g(x) \) and explain where they intersect?
3. What happens if the discriminant of a quadratic equation is negative?
4. How would you solve \( f(x) = g(x) \) if they were higher-degree polynomials?
5. How would this problem differ if one or both functions involved logarithmic terms?

**Tip:** Always check the discriminant \( b^2 - 4ac \) before solving a quadratic, as it tells you whether the solutions are real or complex.

f(x)=x^2-x+2, g(x)=2x^2-3x-6. If f(x) equals g(x), then x equals?

Discover how to solve for x when two quadratic functions f(x) = x^2 - x + 2 and g(x) = 2x^2 - 3x - 6 are set equal. This solution uses the quadratic formula and covers algebraic concepts, suitable for students in Grades 9-11.

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