Use the quadratic formula to solve. Express your answer in simplest form. $\newline$ $3c^{2}+14c-8=4c$

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Algebra 2

Quadratic equation with complex roots

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Q. Use the quadratic formula to solve. Express your answer in simplest form.

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3c^{2}+14c-8=4c

Identify coefficients: Identify the coefficients $a$ , $b$ , and $c$ in the quadratic equation $3c^2 + 10c - 8 = 0$ . Comparing $3c^2 + 10c - 8$ with $ax^2 + bx + c$ , we find $a=3$ , $b=10$ , and $c=-8$ .
Use quadratic formula: Use the quadratic formula to find the roots of the equation. $\newline$ The quadratic formula is $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . $\newline$ Substitute $3$ for $a$ , $10$ for $b$ , and $-8$ for $c$ in the formula. $\newline$ $\frac{-10 \pm \sqrt{10^2 - 4\cdot3\cdot(-8)}}{2\cdot3}$
Simplify terms and calculate: Simplify the terms under the square root. $\newline$ Calculate $10^2$ and $4\times3\times(-8)$ . $\newline$ $10^2 = 100$ $\newline$ $4\times3\times(-8) = -96$ $\newline$ So, $\frac{-10 \pm \sqrt{100 - (-96)}}{2\times3}$
Simplify expression and eliminate: Simplify the expression under the square root. $\newline$ $100 - (-96) = 100 + 96 = 196$ $\newline$ So, $(-10 \pm \sqrt{196})/(2\cdot3)$
Solve for plus and minus: Eliminate the square root and simplify the fraction. $\newline$ $\sqrt{196} = 14$ $\newline$ So, $(-10 \pm 14)/(2\cdot3)$
Write roots in simplest form: Solve the equation for both plus and minus scenarios. $\newline$ $(-10 + 14)/(2*3) = 4/(2*3) = 4/6 = 2/3$ $\newline$ $(-10 - 14)/(2*3) = -24/(2*3) = -24/6 = -4$
Write roots in simplest form: Solve the equation for both plus and minus scenarios. $\newline$ $(-10 + 14)/(2*3) = 4/(2*3) = 4/6 = 2/3$ $\newline$ $(-10 - 14)/(2*3) = -24/(2*3) = -24/6 = -4$ Write the roots in the simplest form. $\newline$ The roots of the equation are $2/3$ and $-4$ .