Solve the equation for all real solutions in simplest form. 2r^2 +3r-5=-1

We are given the equation:

\[
2r^2 + 3r - 5 = -1
\]

### Step 1: Simplify the equation
Start by adding 1 to both sides to move the constant term to the right-hand side:

\[
2r^2 + 3r - 5 + 1 = 0
\]

This simplifies to:

\[
2r^2 + 3r - 4 = 0
\]

### Step 2: Use the quadratic formula
The quadratic formula is:

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

For the quadratic equation \(2r^2 + 3r - 4 = 0\), we identify:
- \(a = 2\)
- \(b = 3\)
- \(c = -4\)

Now, substitute these values into the quadratic formula:

\[
r = \frac{-3 \pm \sqrt{3^2 - 4(2)(-4)}}{2(2)}
\]

### Step 3: Simplify inside the square root
First, calculate the discriminant:

\[
3^2 - 4(2)(-4) = 9 + 32 = 41
\]

Now, substitute the discriminant into the quadratic formula:

\[
r = \frac{-3 \pm \sqrt{41}}{4}
\]

### Step 4: Write the solutions
Thus, the two solutions for \(r\) are:

\[
r = \frac{-3 + \sqrt{41}}{4} \quad \text{and} \quad r = \frac{-3 - \sqrt{41}}{4}
\]

These are the solutions in their simplest form.

---

Would you like more details about the steps? 

Here are some related questions for practice:
1. How do you complete the square for quadratic equations?
2. What happens if the discriminant is negative in the quadratic formula?
3. How do you factor simple quadratic equations?
4. Can you solve quadratic equations by graphing? 
5. What are some common uses of quadratic equations in real-world scenarios?

**Tip:** Always check if you can factor a quadratic equation before using the quadratic formula; it may save you time!

Solve the equation for all real solutions in simplest form. 2r^2 + 3r - 5 = -1

Learn how to solve the quadratic equation 2r^2 + 3r - 5 = -1 in a few easy steps using the quadratic formula. This guide explains the key concepts of algebra, quadratic equations, and discriminants. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation