Solve for x in the equation 2 x squared + 3 x minus 7 = x squared + 5 x + 39.
x = negative 6 plus-or-minus StartRoot 82 EndRoot
x = negative 6 plus-or-minus 2 StartRoot 17 EndRoot
x = 1 plus-or-minus StartRoot 33 EndRoot
x = 1 plus-or-minus StartRoot 47 EndRoot

Let's solve the equation step by step:

The given equation is:

\[
2x^2 + 3x - 7 = x^2 + 5x + 39
\]

### Step 1: Move all terms to one side of the equation
Subtract \(x^2\), \(5x\), and 39 from both sides:

\[
2x^2 + 3x - 7 - (x^2 + 5x + 39) = 0
\]

Simplify:

\[
2x^2 + 3x - 7 - x^2 - 5x - 39 = 0
\]

\[
x^2 - 2x - 46 = 0
\]

### Step 2: Apply the quadratic formula
The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(x^2 - 2x - 46 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -2\)
- \(c = -46\)

Now, substitute these values into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-46)}}{2(1)}
\]

\[
x = \frac{2 \pm \sqrt{4 + 184}}{2}
\]

\[
x = \frac{2 \pm \sqrt{188}}{2}
\]

\[
x = \frac{2 \pm \sqrt{4 \times 47}}{2}
\]

\[
x = \frac{2 \pm 2\sqrt{47}}{2}
\]

\[
x = 1 \pm \sqrt{47}
\]

### Final Answer:
\[
x = 1 \pm \sqrt{47}
\]

Thus, the correct option is:

\[
x = 1 \pm \sqrt{47}
\]

Would you like more details on the quadratic formula or have any questions?

Here are 5 follow-up questions:
1. What is the significance of the discriminant in the quadratic formula?
2. How do you interpret the solutions \(1 + \sqrt{47}\) and \(1 - \sqrt{47}\)?
3. Can you solve a quadratic equation using factoring in some cases instead of the quadratic formula?
4. How does the quadratic formula relate to the graph of a parabola?
5. How would the solution change if the equation had complex roots?

**Tip:** Always check if factoring can simplify a quadratic equation before using the quadratic formula!

Solve for x in the equation 2 x squared + 3 x minus 7 = x squared + 5 x + 39. x = negative 6 plus-or-minus StartRoot 82 EndRoot x = negative 6 plus-or-minus 2 StartRoot 17 EndRoot x = 1 plus-or-minus StartRoot 33 EndRoot x = 1 plus-or-minus StartRoot 47 EndRoot

This tutorial explains how to solve the quadratic equation 2x^2 + 3x - 7 = x^2 + 5x + 39 step-by-step using the quadratic formula. Learn how to simplify the equation and apply the quadratic formula to find solutions involving square roots. Suitable for students in Grades 9-12.

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