Solving the Quadratic Equation $4x^2 + 4x - 3 = 0$

To find the roots of the quadratic equation $4x^2 + 4x - 3 = 0$, we can use the factorization method.

Factorization Steps

1. Identify Coefficients: The equation is in the form $ax^2 + bx + c = 0$, where $a=4$, $b=4$, and $c=-3$.
2. Find Product and Sum: We need two numbers that multiply to $a \times c = 4 \times (-3) = -12$ and add up to $b = 4$. These numbers are $6$ and $-2$.
3. Rewrite Middle Term: Rewrite the equation by splitting the middle term ($4x$) using these numbers:

   $4x^2 + 6x - 2x - 3 = 0$

4. Factor by Grouping: Group the terms and factor out common factors:

   $2x(2x + 3) - 1(2x + 3) = 0$

5. Factor Common Binomial: Factor out the common binomial $(2x + 3)$:

   $(2x - 1)(2x + 3) = 0$

6. Solve for x: Set each factor equal to zero and solve for $x$:
   • $2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}$
   • $2x + 3 = 0 \implies 2x = -3 \implies x = -\frac{3}{2}$

Conclusion

The roots of the equation $4x^2 + 4x - 3 = 0$ are $x = \frac{1}{2}$ and $x = -\frac{3}{2}$.