Solve: \( 4 x^{2}+8 x+1=0 \)

To solve the quadratic equation: \[ 4x^{2} + 8x + 1 = 0 \] we can use the **quadratic formula**, which is given by: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^{2} + bx + c = 0 \). ### Step 1: Identify the coefficients For the equation \( 4x^{2} + 8x + 1 = 0 \): - \( a = 4 \) - \( b = 8 \) - \( c = 1 \) ### Step 2: Compute the discriminant The discriminant (\( \Delta \)) is the part under the square root in the quadratic formula: \[ \Delta = b^{2} - 4ac \] Plugging in the values: \[ \Delta = 8^{2} - 4(4)(1) = 64 - 16 = 48 \] ### Step 3: Apply the quadratic formula Now, substitute \( a \), \( b \), and \( \Delta \) into the quadratic formula: \[ x = \frac{-8 \pm \sqrt{48}}{2 \times 4} = \frac{-8 \pm \sqrt{16 \times 3}}{8} = \frac{-8 \pm 4\sqrt{3}}{8} \] ### Step 4: Simplify the expression Divide numerator and denominator by 4: \[ x = \frac{-2 \pm \sqrt{3}}{2} \] Alternatively, this can be written as: \[ x = -1 \pm \frac{\sqrt{3}}{2} \] ### Final Answer The solutions to the equation \( 4x^{2} + 8x + 1 = 0 \) are: \[ x = -1 + \frac{\sqrt{3}}{2} \quad \text{and} \quad x = -1 - \frac{\sqrt{3}}{2} \] Or, equivalently: \[ x = \frac{-2 \pm \sqrt{3}}{2} \]