Full solution

Q. Solve using the quadratic formula.

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3y^2 - 6y - 4 = 0

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

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y =

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Identify values of $a$ , $b$ , $c$ : Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $3y^2 − 6y − 4 = 0$ . The quadratic equation is in the form $ay^2 + by + c = 0$ . For our equation, $a = 3$ , $b = -6$ , and $b$ $0$ .
Substitute values into quadratic formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The quadratic formula is $y = \frac{-(-6) \pm \sqrt{(-6)^2 - 4\cdot3\cdot(-4)}}{2\cdot3}$ .
Simplify expression inside square root: Simplify the expression inside the square root and the constants outside the square root. $\newline$ Calculate $(-6)^2$ , which is $36$ , and $4 \times 3 \times (-4)$ , which is $-48$ . The expression inside the square root becomes $\sqrt{36 - (-48)}$ .
Further simplify inside square root: Further simplify the expression inside the square root. $\sqrt{36 - (-48)}$ simplifies to $\sqrt{36 + 48}$ which is $\sqrt{84}$ .
Simplify $\sqrt{84}$ : Simplify $\sqrt{84}$ to its simplest radical form. $\sqrt{84}$ can be simplified to $\sqrt{4\times21}$ which is $2\times\sqrt{21}$ .
Substitute simplified square root: Substitute the simplified square root back into the quadratic formula. $\newline$ The quadratic formula now becomes $y = \frac{6 \pm 2\sqrt{21}}{6}$ .
Simplify quadratic formula: Simplify the quadratic formula by dividing all terms by the common factor of $6$ . $y = (1 \pm (\frac{1}{3})\sqrt{21})$ . This gives us two possible solutions for $y$ .
Calculate two possible solutions: Calculate the two possible solutions for $y$ . The first solution is $y = 1 + \frac{1}{3}\sqrt{21}$ , and the second solution is $y = 1 - \frac{1}{3}\sqrt{21}$ .
Round values of y: If necessary, round the values of $y$ to the nearest hundredth. $y \approx 1 + 0.333 \times 4.58$ or $y \approx 1 - 0.333 \times 4.58$ $y \approx 1 + 1.53$ or $y \approx 1 - 1.53$ $y \approx 2.53$ or $y \approx -0.53$