Solve using the quadratic formula. $\newline$ $6z^2 + 8z - 7 = 0$ $\newline$ 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

6z^2 + 8z - 7 = 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 quadratic equation $az^2 + bz + c = 0$ . In this case, $a = 6$ , $b = 8$ , and $c = -7$ .
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 $8^2 - 4(6)(-7)$ .
Apply Quadratic Formula: Perform the calculation: $64 - (-168) = 64 + 168 = 232$ . The discriminant is $232$ , which is positive, indicating that there are two real solutions.
Calculate z with Plus: Now, apply the quadratic formula using the values of $a$ , $b$ , and $c$ : $z = \frac{-8 \pm \sqrt{232}}{2 \times 6}$ .
Calculate z with Minus: Simplify the formula: $z = \frac{{-8 \pm \sqrt{232}}}{{12}}$ .
Calculate z with Minus: Simplify the formula: $z = \frac{-8 \pm \sqrt{232}}{12}$ .Calculate the two possible values for z by considering the plus and minus in the formula separately. First, calculate z when using the plus: $z = \frac{-8 + \sqrt{232}}{12}$ .
Calculate z with Minus: Simplify the formula: $z = \frac{-8 \pm \sqrt{232}}{12}$ .Calculate the two possible values for z by considering the plus and minus in the formula separately. First, calculate z when using the plus: $z = \frac{-8 + \sqrt{232}}{12}$ .Using a calculator, find the square root of $232$ , which is approximately $15.23$ (rounded to the nearest hundredth).
Calculate z with Minus: Simplify the formula: $z = \frac{-8 \pm \sqrt{232}}{12}$ .Calculate the two possible values for z by considering the plus and minus in the formula separately. First, calculate z when using the plus: $z = \frac{-8 + \sqrt{232}}{12}$ .Using a calculator, find the square root of $232$ , which is approximately $15.23$ (rounded to the nearest hundredth).Now, substitute $\sqrt{232}$ with $15.23$ in the equation: $z = \frac{-8 + 15.23}{12}$ .
Calculate z with Minus: Simplify the formula: $z = \frac{-8 \pm \sqrt{232}}{12}$ .Calculate the two possible values for z by considering the plus and minus in the formula separately. First, calculate z when using the plus: $z = \frac{-8 + \sqrt{232}}{12}$ .Using a calculator, find the square root of $232$ , which is approximately $15.23$ (rounded to the nearest hundredth).Now, substitute $\sqrt{232}$ with $15.23$ in the equation: $z = \frac{-8 + 15.23}{12}$ .Perform the calculation: $z = \frac{7.23}{12} \approx 0.60$ (rounded to the nearest hundredth).
Calculate z with Minus: Simplify the formula: $z = \frac{-8 \pm \sqrt{232}}{12}$ .Calculate the two possible values for z by considering the plus and minus in the formula separately. First, calculate z when using the plus: $z = \frac{-8 + \sqrt{232}}{12}$ .Using a calculator, find the square root of $232$ , which is approximately $15.23$ (rounded to the nearest hundredth).Now, substitute $\sqrt{232}$ with $15.23$ in the equation: $z = \frac{-8 + 15.23}{12}$ .Perform the calculation: $z = \frac{7.23}{12} \approx 0.60$ (rounded to the nearest hundredth).Next, calculate z when using the minus: $z = \frac{-8 - \sqrt{232}}{12}$ .
Calculate z with Minus: Simplify the formula: $z = \frac{{-8 \pm \sqrt{232}}}{{12}}$ .Calculate the two possible values for z by considering the plus and minus in the formula separately. First, calculate z when using the plus: $z = \frac{{-8 + \sqrt{232}}}{{12}}$ .Using a calculator, find the square root of $232$ , which is approximately $15.23$ (rounded to the nearest hundredth).Now, substitute $\sqrt{232}$ with $15.23$ in the equation: $z = \frac{{-8 + 15.23}}{{12}}$ .Perform the calculation: $z = \frac{{7.23}}{{12}} \approx 0.60$ (rounded to the nearest hundredth).Next, calculate z when using the minus: $z = \frac{{-8 - \sqrt{232}}}{{12}}$ .Substitute $\sqrt{232}$ with $15.23$ in the equation: $z = \frac{{-8 + \sqrt{232}}}{{12}}$ $0$ .
Calculate z with Minus: Simplify the formula: $z = \frac{-8 \pm \sqrt{232}}{12}$ .Calculate the two possible values for z by considering the plus and minus in the formula separately. First, calculate z when using the plus: $z = \frac{-8 + \sqrt{232}}{12}$ .Using a calculator, find the square root of $232$ , which is approximately $15.23$ (rounded to the nearest hundredth).Now, substitute $\sqrt{232}$ with $15.23$ in the equation: $z = \frac{-8 + 15.23}{12}$ .Perform the calculation: $z = \frac{7.23}{12} \approx 0.60$ (rounded to the nearest hundredth).Next, calculate z when using the minus: $z = \frac{-8 - \sqrt{232}}{12}$ .Substitute $\sqrt{232}$ with $15.23$ in the equation: $z = \frac{-8 + \sqrt{232}}{12}$ $0$ .Perform the calculation: $z = \frac{-8 + \sqrt{232}}{12}$ $1$ (rounded to the nearest hundredth).