Full solution

Q. Solve using the quadratic formula.

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4x^2 + 8x + 1 = 0

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Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

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x =

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x =

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Identify coefficients: To solve the quadratic equation $4x^2 + 8x + 1 = 0$ using the quadratic formula, we first identify the coefficients $a$ , $b$ , and $c$ from the standard form of a quadratic equation $ax^2 + bx + c = 0$ . Here, $a = 4$ , $b = 8$ , and $c = 1$ .
Apply quadratic formula: The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . We will substitute the values of $a$ , $b$ , and $c$ into this formula to find the solutions for $x$ .
Calculate discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ . For our equation, the discriminant is $8^2 - 4(4)(1) = 64 - 16 = 48$ .
Substitute values: Now, we can substitute the values into the quadratic formula: $x = \frac{-8 \pm \sqrt{48}}{2 \times 4}$ . This simplifies to $x = \frac{-8 \pm \sqrt{48}}{8}$ .
Simplify square root: We can simplify $\sqrt{48}$ by factoring it into $\sqrt{(16 \times 3)}$ , which is $4\sqrt{3}$ . So the equation becomes $x = \frac{(-8 \pm 4\sqrt{3})}{8}$ .
Divide by $8$ : We can now simplify the equation further by dividing both terms in the numerator by $8$ : $x = (-1 \pm \sqrt{3}/2)$ .
Find two solutions: This gives us two solutions for $x$ : $x = -1 + \sqrt{3}/2$ and $x = -1 - \sqrt{3}/2$ . These are the solutions in their simplest radical form. If we need decimal approximations, we can calculate these values.
Calculate decimal approximations: Calculating the decimal approximations: $x \approx -1 + (\frac{1.732}{2}) \approx -1 + 0.866 \approx -0.134$ and $x \approx -1 - (\frac{1.732}{2}) \approx -1 - 0.866 \approx -1.866$ . Rounding to the nearest hundredth, we get $x \approx -0.13$ and $x \approx -1.87$ .