Salio in quadratic oquation by facturizing \( 4 x^{2}+4 x-3=0 \)

Solve the quadratic equation by following steps: - step0: Solve by factoring: \(4x^{2}+4x-3=0\) - step1: Factor the expression: \(\left(2x-1\right)\left(2x+3\right)=0\) - step2: Separate into possible cases: \(\begin{align}&2x-1=0\\&2x+3=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=\frac{1}{2}\\&x=-\frac{3}{2}\end{align}\) - step4: Rewrite: \(x_{1}=-\frac{3}{2},x_{2}=\frac{1}{2}\) To solve the quadratic equation \( 4x^{2} + 4x - 3 = 0 \) by factoring, we can follow these steps: 1. **Identify the coefficients**: In the equation \( ax^2 + bx + c = 0 \), we have: - \( a = 4 \) - \( b = 4 \) - \( c = -3 \) 2. **Factor the quadratic**: We need to find two numbers that multiply to \( ac = 4 \times -3 = -12 \) and add to \( b = 4 \). The numbers that satisfy this are \( 6 \) and \( -2 \). 3. **Rewrite the equation**: We can rewrite the middle term using these numbers: \[ 4x^{2} + 6x - 2x - 3 = 0 \] 4. **Group the terms**: Now, we group the terms: \[ (4x^{2} + 6x) + (-2x - 3) = 0 \] 5. **Factor by grouping**: \[ 2x(2x + 3) - 1(2x + 3) = 0 \] This can be factored as: \[ (2x + 3)(2x - 1) = 0 \] 6. **Set each factor to zero**: \[ 2x + 3 = 0 \quad \text{or} \quad 2x - 1 = 0 \] 7. **Solve for \( x \)**: - From \( 2x + 3 = 0 \): \[ 2x = -3 \implies x = -\frac{3}{2} \] - From \( 2x - 1 = 0 \): \[ 2x = 1 \implies x = \frac{1}{2} \] Thus, the solutions to the equation \( 4x^{2} + 4x - 3 = 0 \) are: \[ x_{1} = -\frac{3}{2}, \quad x_{2} = \frac{1}{2} \]