To solve the quadratic equation [tex]\( 4x^2 + 4x - 3 = 0 \)[/tex] for the values of [tex]\( x \)[/tex], let's follow a step-by-step approach.
### Step 1: Identify the coefficients
For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we have:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = -3 \)[/tex]
### Step 2: Use the Quadratic Formula
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula
Plugging in the values, we get:
[tex]\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 4 \cdot (-3)}}{2 \cdot 4} \][/tex]
### Step 4: Simplify under the square root
First, calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac = 4^2 - 4 \cdot 4 \cdot (-3) = 16 + 48 = 64 \][/tex]
The discriminant is 64, which is a perfect square.
### Step 5: Continue simplifying the equation
Now we substitute the discriminant back into the formula:
[tex]\[ x = \frac{-4 \pm \sqrt{64}}{8} \][/tex]
It simplifies further:
[tex]\[ x = \frac{-4 \pm 8}{8} \][/tex]
### Step 6: Solve for the two potential values of [tex]\( x \)[/tex]
We get two solutions from the [tex]\( \pm \)[/tex] sign:
1. [tex]\( x = \frac{-4 + 8}{8} = \frac{4}{8} = \frac{1}{2} \)[/tex]
2. [tex]\( x = \frac{-4 - 8}{8} = \frac{-12}{8} = \frac{-3}{2} \)[/tex]
### Final Step: Present the solutions
Thus, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 4x^2 + 4x - 3 = 0 \)[/tex] are:
[tex]\[ x = \frac{1}{2} \quad \text{and} \quad x = \frac{-3}{2} \][/tex]
The correct choice among the given options is:
[tex]\[ \boxed{x = -1.5, \, 0.5} \][/tex]
