Solve using the quadratic formula. $\newline$ $8z^2 + 4z - 8 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $z =$ _ or $z =$ _

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Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

8z^2 + 4z - 8 = 0

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

\newline

z =

_____ or

z =

_____

Quadratic Formula: The quadratic formula is given by $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the terms in the quadratic equation $az^2 + bz + c = 0$ . For the equation $8z^2 + 4z - 8 = 0$ , $a = 8$ , $b = 4$ , and $c = -8$ .
Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ . Here, it is $4^2 - 4(8)(-8)$ .
Discriminant Calculation: Perform the calculation: $16 - 4(8)(-8) = 16 + 256 = 272$ .
Insert Values into Formula: Now, insert the values of $a$ , $b$ , and the discriminant into the quadratic formula: $z = \frac{-4 \pm \sqrt{272}}{2 \times 8}$ .
Simplify Formula: Simplify the formula: $z = \frac{{-4 \pm \sqrt{272}}}{{16}}$ .
Find Perfect Square Factors: Since $\sqrt{272}$ is not a perfect square, we can simplify it by looking for perfect square factors. The largest perfect square factor of $272$ is $16$ , so $\sqrt{272} = \sqrt{(16 \times 17)} = 4\sqrt{17}$ .
Replace Square Root: Replace $\sqrt{272}$ with $4\sqrt{17}$ in the formula: $z = \frac{(-4 \pm 4\sqrt{17})}{16}$ .
Factor Out Common Factor: Factor out the common factor of $4$ in the numerator: $z = \frac{4(-1 \pm \sqrt{17})}{16}$ .
Simplify Fraction: Simplify the fraction by dividing both the numerator and the denominator by $4$ : $z = \frac{-1 \pm \sqrt{17}}{4}$ .
Calculate Solutions: Now we have two solutions for $z$ , one using the plus sign and one using the minus sign: $z = \frac{-1 + \sqrt{17}}{4}$ or $z = \frac{-1 - \sqrt{17}}{4}$ .
Express as Decimals: To express the solutions as decimals rounded to the nearest hundredth, calculate each one: $z \approx (-1 + 4.1231) / 4 \approx 3.1231 / 4 \approx 0.78$ and $z \approx (-1 - 4.1231) / 4 \approx -5.1231 / 4 \approx -1.28$ .