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DeltaMath Student Application $\newline$ Desmos | Scientlific Calculator $\newline$ ident/ $3345436$ / $22863483$ /a $11367$ fcoeb $6$ e $689$ ef $6035$ efcaof $83$ a $8$ $\newline$ chool stuff $\newline$ youtube $\newline$ codehs $\newline$ C points $\newline$ ReadTheory | Readi... $\newline$ Space huggers - Op. $\newline$ Score: $0 / 20$ $\newline$ Penalty: $1$ off $\newline$ Question $\newline$ Solve the equation for all real solutions in simplest form. $\newline$ $3 z^{2}-z+5=6$ $\newline$ Answer Attempt y out of $2$ $\newline$ ( $9$ ) Additional Solution $\newline$ No Solution $\newline$ $z=$ $\newline$ Submit Answer

Full solution

Q. DeltaMath Student Application

\newline

Desmos | Scientlific Calculator

\newline

ident/

3345436

22863483

11367

fcoeb

6

689

6035

efcaof

83

8

\newline

chool stuff

\newline

youtube

\newline

codehs

\newline

C points

\newline

ReadTheory | Readi...

\newline

Space huggers - Op.

\newline

Score:

0 / 20

\newline

Penalty:

1

off

\newline

Question

\newline

Solve the equation for all real solutions in simplest form.

\newline

3 z^{2}-z+5=6

\newline

Answer Attempt y out of

2

\newline

(

9

) Additional Solution

\newline

No Solution

\newline

z=

\newline

Submit Answer

Set Equation to Zero: Set the equation to zero by subtracting $6$ from both sides. $\newline$ $3z^2 - z + 5 - 6 = 0$ $\newline$ This simplifies to: $\newline$ $3z^2 - z - 1 = 0$
Factor Quadratic Equation: Try to factor the quadratic equation. $\newline$ We are looking for two numbers that multiply to $(3)(-1) = -3$ and add to $-1$ . However, there are no two integers that satisfy these conditions. Therefore, we cannot factor this quadratic equation easily.
Use Quadratic Formula: Use the quadratic formula to find the solutions for $z$ . The quadratic formula is $z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 3$ , $b = -1$ , and $c = -1$ .
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots. $\newline$ Discriminant = $(-1)^2 - 4(3)(-1) = 1 + 12 = 13$ $\newline$ Since the discriminant is positive, we have two distinct real solutions.
Substitute Values: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula. $\newline$ $z = \frac{-(-1) \pm \sqrt{13}}{2 \times 3}$ $\newline$ This simplifies to: $\newline$ $z = \frac{1 \pm \sqrt{13}}{6}$
Write Solutions: Write down the two solutions. $\newline$ z = $(1 + \sqrt{13}) / 6$ or z = $(1 - \sqrt{13}) / 6$ $\newline$ These are the solutions in simplest form.