Solve using the quadratic formula. $\newline$ $6z^2 - 2z - 7 = 0$ $\newline$ 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 - 2z - 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 terms in the quadratic equation $az^2 + bz + c = 0$ . In this case, $a = 6$ , $b = -2$ , and $c = -7$ .
Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ . Here, it is $(-2)^2 - 4(6)(-7)$ .
Perform Calculation: Perform the calculation: $4 + 168 = 172$ .
Insert Values into Formula: Now, insert the values of $a$ , $b$ , and the discriminant into the quadratic formula to find the two possible values for $z$ . $z = \frac{-(-2) \pm \sqrt{172}}{2 \times 6}$
Simplify Equation: Simplify the equation by calculating the numerator and denominator separately. $z = \frac{2 \pm \sqrt{172}}{12}$
Factor Out Square Root: Since $\sqrt{172}$ is not a perfect square, we can simplify it by factoring out perfect squares. $\sqrt{172}$ can be written as $\sqrt{(4\times43)}$ , which simplifies to $2\sqrt{43}$ .
Substitute Simplified Root: Substitute the simplified square root back into the equation. $z = \frac{2 \pm 2\sqrt{43}}{12}$
Simplify Fraction: Now, we can simplify the fraction by dividing both terms in the numerator by $2$ . $z = \frac{1 \pm \sqrt{43}}{6}$
Find Solutions for $z$ : We have two solutions for $z$ , which are $z = \frac{1 + \sqrt{43}}{6}$ and $z = \frac{1 - \sqrt{43}}{6}$ . These cannot be simplified further into integers or fractions, so we can express them as decimals rounded to the nearest hundredth.
Calculate Decimal Values: Calculate the decimal values for both solutions. $\newline$ $z = \frac{1 + \sqrt{43}}{6} \approx \frac{1 + 6.56}{6} \approx \frac{7.56}{6} \approx 1.26$ $\newline$ $z = \frac{1 - \sqrt{43}}{6} \approx \frac{1 - 6.56}{6} \approx \frac{-5.56}{6} \approx -0.93$