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Solve using the quadratic formula.\newline4x2+8x+1=04x^2 + 8x + 1 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____

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Q. Solve using the quadratic formula.\newline4x2+8x+1=04x^2 + 8x + 1 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____
  1. Identify coefficients: To solve the quadratic equation 4x2+8x+1=04x^2 + 8x + 1 = 0 using the quadratic formula, we first identify the coefficients aa, bb, and cc from the standard form of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. Here, a=4a = 4, b=8b = 8, and c=1c = 1.
  2. Apply quadratic formula: The quadratic formula is given by x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. We will substitute the values of aa, bb, and cc into this formula to find the solutions for xx.
  3. Calculate discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. For our equation, the discriminant is 82−4(4)(1)=64−16=488^2 - 4(4)(1) = 64 - 16 = 48.
  4. Substitute values: Now, we can substitute the values into the quadratic formula: x=−8±482×4x = \frac{-8 \pm \sqrt{48}}{2 \times 4}. This simplifies to x=−8±488x = \frac{-8 \pm \sqrt{48}}{8}.
  5. Simplify square root: We can simplify 48\sqrt{48} by factoring it into (16×3)\sqrt{(16 \times 3)}, which is 434\sqrt{3}. So the equation becomes x=(−8±43)8x = \frac{(-8 \pm 4\sqrt{3})}{8}.
  6. Divide by 88: We can now simplify the equation further by dividing both terms in the numerator by 88: x=(−1±3/2)x = (-1 \pm \sqrt{3}/2).
  7. Find two solutions: This gives us two solutions for xx: x=−1+3/2x = -1 + \sqrt{3}/2 and x=−1−3/2x = -1 - \sqrt{3}/2. These are the solutions in their simplest radical form. If we need decimal approximations, we can calculate these values.
  8. Calculate decimal approximations: Calculating the decimal approximations: x≈−1+(1.7322)≈−1+0.866≈−0.134x \approx -1 + (\frac{1.732}{2}) \approx -1 + 0.866 \approx -0.134 and x≈−1−(1.7322)≈−1−0.866≈−1.866x \approx -1 - (\frac{1.732}{2}) \approx -1 - 0.866 \approx -1.866. Rounding to the nearest hundredth, we get x≈−0.13x \approx -0.13 and x≈−1.87x \approx -1.87.

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