Solve the equation by using the Quadratic Formula. Round to the nearest tenth, if necessary. Write your solutions from least to greatest, separated by a comma, if necessary. If there are no real solutions, write no solutions.

 

−9x2–2x+3=0
 

 

x = __

We are given the quadratic equation:

\[
-9x^2 - 2x + 3 = 0
\]

We will solve this using the **Quadratic Formula**, which is:

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

In this equation, \( a = -9 \), \( b = -2 \), and \( c = 3 \).

### Step 1: Compute the discriminant
The discriminant is given by \( \Delta = b^2 - 4ac \).

\[
\Delta = (-2)^2 - 4(-9)(3) = 4 + 108 = 112
\]

### Step 2: Apply the quadratic formula
Now that we have the discriminant (\( \Delta = 112 \)), we can plug the values into the formula:

\[
x = \frac{-(-2) \pm \sqrt{112}}{2(-9)} = \frac{2 \pm \sqrt{112}}{-18}
\]

### Step 3: Simplify the square root of the discriminant
We know \( \sqrt{112} \approx 10.583 \).

So, the solutions become:

\[
x = \frac{2 \pm 10.583}{-18}
\]

### Step 4: Calculate the two solutions
1. \( x_1 = \frac{2 - 10.583}{-18} = \frac{-8.583}{-18} \approx 0.477 \)
2. \( x_2 = \frac{2 + 10.583}{-18} = \frac{12.583}{-18} \approx -0.699 \)

### Final Answer:
The solutions, from least to greatest, are:

\[
x \approx -0.7, 0.5
\]

Let me know if you need further details or have any questions!

---

**Related questions:**
1. What is the significance of the discriminant in determining the nature of the roots?
2. How can the Quadratic Formula be used to find complex solutions when the discriminant is negative?
3. Can the quadratic equation be factored in cases where the discriminant is a perfect square?
4. What happens when the discriminant equals zero in the quadratic formula?
5. How do changes in the coefficients \( a \), \( b \), and \( c \) affect the graph of a quadratic equation?

**Tip:** Always check the discriminant first to determine whether the solutions will be real or complex!

Solve the equation by using the Quadratic Formula. Round to the nearest tenth, if necessary. Write your solutions from least to greatest, separated by a comma, if necessary. If there are no real solutions, write no solutions.

−9x^2–2x+3=0

x = __

Learn how to solve the quadratic equation -9x^2 - 2x + 3 using the quadratic formula. Follow a detailed step-by-step guide to find the solutions, rounded to the nearest tenth. This tutorial covers algebraic concepts such as the discriminant and the quadratic formula, suitable for students in Grades 9-11.

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