4(x² - 7) = x(x - 2) transform the equation into quadratic equation and find the solution set

Let's start by expanding both sides of the equation and then rearranging the terms to form a quadratic equation.

Given equation:
\[
4(x^2 - 7) = x(x - 2)
\]

### Step 1: Expand both sides
Expand the left side:
\[
4(x^2 - 7) = 4x^2 - 28
\]

Expand the right side:
\[
x(x - 2) = x^2 - 2x
\]

Now the equation is:
\[
4x^2 - 28 = x^2 - 2x
\]

### Step 2: Bring all terms to one side to form a quadratic equation
Subtract \(x^2\) and add \(2x\) to both sides:
\[
4x^2 - 28 - x^2 + 2x = 0
\]

Simplify the equation:
\[
3x^2 + 2x - 28 = 0
\]

### Step 3: Solve the quadratic equation
The quadratic equation is:
\[
3x^2 + 2x - 28 = 0
\]

To solve for \(x\), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \(a = 3\), \(b = 2\), and \(c = -28\).

### Step 4: Calculate the discriminant
\[
\text{Discriminant} = b^2 - 4ac = 2^2 - 4(3)(-28) = 4 + 336 = 340
\]

### Step 5: Find the solutions
\[
x = \frac{-2 \pm \sqrt{340}}{6}
\]

Simplifying the square root of 340:
\[
\sqrt{340} = \sqrt{4 \times 85} = 2\sqrt{85}
\]
Thus,
\[
x = \frac{-2 \pm 2\sqrt{85}}{6} = \frac{-1 \pm \sqrt{85}}{3}
\]

### Solution Set:
The solutions for \(x\) are:
\[
x_1 = \frac{-1 + \sqrt{85}}{3}, \quad x_2 = \frac{-1 - \sqrt{85}}{3}
\]

Do you want more details or have any questions?

---

**Related Questions:**

1. How can we simplify the quadratic formula further for specific cases?
2. What is the significance of the discriminant in solving quadratic equations?
3. Can quadratic equations have complex solutions?
4. How can you verify the roots of a quadratic equation?
5. What are some applications of quadratic equations in real-world problems?

**Tip:** Always check the discriminant first to understand the nature of the roots before solving the quadratic equation.

Learn how to transform and solve the quadratic equation 4(x² - 7) = x(x - 2). This detailed step-by-step solution covers expanding equations, applying the quadratic formula, and calculating the discriminant. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation