what is the quadratic formula of 2x²+9x=x-7 

To solve the quadratic equation \(2x^2 + 9x = x - 7\) using the quadratic formula, follow these steps:

1. **Rewrite the equation in standard form \(ax^2 + bx + c = 0\):**
   \[
   2x^2 + 9x - x + 7 = 0
   \]
   Simplify the equation:
   \[
   2x^2 + 8x + 7 = 0
   \]

2. **Identify the coefficients \(a\), \(b\), and \(c\):**
   \[
   a = 2, \quad b = 8, \quad c = 7
   \]

3. **Write down the quadratic formula:**
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

4. **Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula:**
   \[
   x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 2 \cdot 7}}{2 \cdot 2}
   \]

5. **Simplify inside the square root:**
   \[
   x = \frac{-8 \pm \sqrt{64 - 56}}{4}
   \]
   \[
   x = \frac{-8 \pm \sqrt{8}}{4}
   \]

6. **Simplify the square root and the expression:**
   \[
   x = \frac{-8 \pm 2\sqrt{2}}{4}
   \]
   \[
   x = \frac{-8}{4} \pm \frac{2\sqrt{2}}{4}
   \]
   \[
   x = -2 \pm \frac{\sqrt{2}}{2}
   \]

So, the solutions to the equation \(2x^2 + 9x = x - 7\) using the quadratic formula are:
\[
x = -2 + \frac{\sqrt{2}}{2} \quad \text{and} \quad x = -2 - \frac{\sqrt{2}}{2}
\]

Would you like further details or have any questions?

Here are some related questions for further practice:

1. Use the quadratic formula to solve \(3x^2 + 5x - 2 = 0\).
2. Use the quadratic formula to solve \(x^2 + 4x + 4 = 0\).
3. Use the quadratic formula to solve \(4x^2 - 12x + 9 = 0\).
4. Use the quadratic formula to solve \(x^2 - 6x + 9 = 0\).
5. Use the quadratic formula to solve \(5x^2 + 3x - 2 = 0\).
6. Use the quadratic formula to solve \(6x^2 + x - 2 = 0\).
7. Use the quadratic formula to solve \(x^2 + 7x + 10 = 0\).
8. Use the quadratic formula to solve \(2x^2 - 5x + 3 = 0\).

**Tip:** Always simplify the expression inside the square root first before proceeding with the rest of the quadratic formula.

Learn how to solve the quadratic equation 2x² + 9x = x - 7 step-by-step using the quadratic formula. This problem covers key algebraic concepts, including the quadratic formula and standard form equations. Suitable for high school students.

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