Let's solve the equation step by step:

### Given equation:
\[
5x^2 - 13x + 15 = -5x
\]

### Step 1: Bring all terms to one side
We can move \(-5x\) from the right side to the left side by adding \(5x\) to both sides:
\[
5x^2 - 13x + 15 + 5x = 0
\]
This simplifies to:
\[
5x^2 - 8x + 15 = 0
\]

### Step 2: Check the discriminant
This is a quadratic equation in the form \(ax^2 + bx + c = 0\). The coefficients are:
- \(a = 5\)
- \(b = -8\)
- \(c = 15\)

To determine whether this equation has real solutions, we calculate the discriminant \(\Delta\):
\[
\Delta = b^2 - 4ac = (-8)^2 - 4(5)(15) = 64 - 300 = -236
\]

### Step 3: Interpret the discriminant
Since the discriminant is negative (\(\Delta = -236\)), the quadratic equation has **no real solutions**. It has **two complex solutions**.

Do you need further details on solving the equation in the complex number field?

---

Here are five related questions:
1. What is the discriminant and how does it determine the number of solutions?
2. How do you solve a quadratic equation with a negative discriminant?
3. What are complex numbers, and how do they relate to quadratic equations?
4. How can you factor quadratic equations when real solutions exist?
5. How do quadratic equations apply to real-world problems?

**Tip:** The discriminant (\(b^2 - 4ac\)) helps determine if a quadratic equation has real or complex solutions. If \(\Delta > 0\), there are two real solutions; if \(\Delta = 0\), there is one real solution; and if \(\Delta < 0\), there are two complex solutions.

Learn how to solve the quadratic equation 5x^2 - 13x + 15 = -5x with a detailed, step-by-step explanation. This problem involves key algebraic concepts, such as the discriminant and complex numbers. Suitable for high school students, grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation