hi! im having trouble understanding this equations and id love your help! 5x^2 - 4x = 6

Certainly! Let's solve the equation \( 5x^2 - 4x = 6 \).

### Step 1: Move all terms to one side of the equation.
We start by subtracting 6 from both sides to set the equation equal to zero:
\[
5x^2 - 4x - 6 = 0
\]

### Step 2: Apply the quadratic formula.
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 5 \), \( b = -4 \), and \( c = -6 \).

### Step 3: Calculate the discriminant.
The discriminant \( \Delta \) is the part under the square root:
\[
\Delta = b^2 - 4ac = (-4)^2 - 4(5)(-6)
\]
\[
\Delta = 16 + 120 = 136
\]

### Step 4: Find the roots.
Now plug the values into the quadratic formula:
\[
x = \frac{-(-4) \pm \sqrt{136}}{2(5)}
\]
\[
x = \frac{4 \pm \sqrt{136}}{10}
\]
Since \( \sqrt{136} \) simplifies to \( 2\sqrt{34} \), we get:
\[
x = \frac{4 \pm 2\sqrt{34}}{10}
\]
Simplifying further:
\[
x = \frac{2 \pm \sqrt{34}}{5}
\]

### Final Answer:
The solutions to the equation are:
\[
x = \frac{2 + \sqrt{34}}{5} \quad \text{or} \quad x = \frac{2 - \sqrt{34}}{5}
\]

Would you like further details on any specific part of the process or additional explanations?

Here are some related questions you might find useful:
1. How do you simplify square roots like \(\sqrt{136}\)?
2. What is the significance of the discriminant in the quadratic formula?
3. How do you factor a quadratic equation when the quadratic formula isn't needed?
4. Can quadratic equations have complex solutions?
5. How do you graph quadratic equations to find their roots?

**Tip:** Always check your solutions by plugging them back into the original equation to verify they satisfy it!

Learn how to solve the quadratic equation 5x^2 - 4x = 6 with a detailed step-by-step explanation. This problem involves key algebraic concepts such as applying the quadratic formula and simplifying square roots. 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