To solve the quadratic equation \( 4x^2 - 8x = 3 \), follow these steps:

### Step 1: Move all terms to one side
We want to write the equation in the standard form \( ax^2 + bx + c = 0 \). So, subtract 3 from both sides:

\[
4x^2 - 8x - 3 = 0
\]

### Step 2: Apply the quadratic formula
The quadratic formula is given by:

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

For the equation \( 4x^2 - 8x - 3 = 0 \), the coefficients are:
- \( a = 4 \)
- \( b = -8 \)
- \( c = -3 \)

### Step 3: Calculate the discriminant
The discriminant is the part under the square root in the quadratic formula, \( b^2 - 4ac \):

\[
\text{Discriminant} = (-8)^2 - 4(4)(-3) = 64 + 48 = 112
\]

### Step 4: Solve for \( x \)
Now plug the values into the quadratic formula:

\[
x = \frac{-(-8) \pm \sqrt{112}}{2(4)}
\]

\[
x = \frac{8 \pm \sqrt{112}}{8}
\]

Since \( \sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7} \), we can simplify further:

\[
x = \frac{8 \pm 4\sqrt{7}}{8}
\]

Now, simplify the expression:

\[
x = \frac{8}{8} \pm \frac{4\sqrt{7}}{8}
\]

\[
x = 1 \pm \frac{\sqrt{7}}{2}
\]

### Final answer:
The two solutions are:

\[
x = 1 + \frac{\sqrt{7}}{2} \quad \text{or} \quad x = 1 - \frac{\sqrt{7}}{2}
\]

Would you like more details on any of these steps?

Here are five related questions for practice:
1. Solve \( 5x^2 - 10x + 2 = 0 \) using the quadratic formula.
2. Factor the quadratic equation \( 2x^2 - 3x - 5 = 0 \).
3. What is the discriminant of \( x^2 + 4x + 4 = 0 \)?
4. Solve \( 3x^2 - 12x + 9 = 0 \) by completing the square.
5. What are the solutions to \( x^2 + 2x - 15 = 0 \) by factoring?

**Tip:** The discriminant \( b^2 - 4ac \) helps you determine the nature of the roots (real, complex, or repeated).

Learn how to solve the quadratic equation 4x^2 - 8x = 3 step-by-step using the quadratic formula. This problem involves calculating the discriminant and simplifying expressions. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation