Solve using the quadratic formula. $\newline$ $8z^2 + 2z - 9 = 0$ $\newline$ 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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Algebra 2

Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

8z^2 + 2z - 9 = 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 =

_____

Identify Coefficients: Identify the coefficients of the quadratic equation. $\newline$ The quadratic equation is in the form $az^2 + bz + c = 0$ . For the equation $8z^2 + 2z - 9 = 0$ , the coefficients are: $\newline$ a = $8$ $\newline$ b = $2$ $\newline$ c = $-9$
Substitute into Formula: Substitute the coefficients into the quadratic formula. $\newline$ The quadratic formula is $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substituting the values of $a$ , $b$ , and $c$ , we get: $\newline$ $z = \frac{-(2) \pm \sqrt{(2)^2 - 4(8)(-9)}}{2(8)}$
Simplify Square Root: Simplify under the square root. $\newline$ Calculate the discriminant (the expression under the square root): $\newline$ Discriminant = $(2)^2 - 4(8)(-9)$ $\newline$ Discriminant = $4 + 288$ $\newline$ Discriminant = $292$
Continue with Formula: Continue with the quadratic formula. $\newline$ Now we have the discriminant, we can continue with the quadratic formula: $\newline$ $z = \frac{-2 \pm \sqrt{292}}{16}$
Simplify Solutions: Simplify the solutions. $\newline$ We have two possible solutions for $z$ , corresponding to the ' $\pm$ ' in the formula: $\newline$ $z = \frac{-2 + \sqrt{292}}{16}$ or $z = \frac{-2 - \sqrt{292}}{16}$
Calculate Numerical Values: Calculate the numerical values and round to the nearest hundredth if necessary. $\newline$ First, calculate the square root of $292$ : $\newline$ $\sqrt{292} \approx 17.09$ $\newline$ Now, substitute this value into the solutions: $\newline$ $z = \frac{-2 + 17.09}{16}$ or $z = \frac{-2 - 17.09}{16}$ $\newline$ $z \approx \frac{15.09}{16}$ or $z \approx \frac{-19.09}{16}$ $\newline$ $z \approx 0.94$ or $z \approx -1.19$