https://file.aiquickdraw.com/mathbot/file/1723916375238hlakhn3a.jpg

The equation provided in the image is:

\[
7x^2 + 10x = 6
\]

To find the solutions, we'll start by rewriting the equation in standard quadratic form \(ax^2 + bx + c = 0\). 

First, subtract 6 from both sides:

\[
7x^2 + 10x - 6 = 0
\]

Now, the quadratic equation is in standard form:

\[
7x^2 + 10x - 6 = 0
\]

We can solve this quadratic equation using the quadratic formula:

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

For this equation:

- \(a = 7\)
- \(b = 10\)
- \(c = -6\)

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula:

\[
x = \frac{-10 \pm \sqrt{10^2 - 4(7)(-6)}}{2(7)}
\]

Simplify under the square root:

\[
x = \frac{-10 \pm \sqrt{100 + 168}}{14}
\]

\[
x = \frac{-10 \pm \sqrt{268}}{14}
\]

Now, simplify the square root of 268:

\[
\sqrt{268} = \sqrt{4 \times 67} = 2\sqrt{67}
\]

So the equation becomes:

\[
x = \frac{-10 \pm 2\sqrt{67}}{14}
\]

This simplifies to:

\[
x = \frac{-5 \pm \sqrt{67}}{7}
\]

Thus, the two solutions are:

- Smaller solution: \( x = \frac{-5 - \sqrt{67}}{7} \)
- Larger solution: \( x = \frac{-5 + \sqrt{67}}{7} \)

Would you like the exact decimal values for these solutions, or is this form sufficient?

Here are 5 related questions to expand on this topic:

1. How do you determine the discriminant of a quadratic equation and what does it signify?
2. Can the quadratic formula be used for any quadratic equation?
3. What are some alternative methods to solve quadratic equations?
4. How do you graphically interpret the solutions of a quadratic equation?
5. What happens if the discriminant is zero or negative?

**Tip:** When solving quadratic equations, always check if the quadratic can be factored easily before applying the quadratic formula. It might save time!

Learn how to solve the quadratic equation 7x^2 + 10x - 6 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and discriminant. Suitable for students in Grades 10-12.

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