Step 1a

Case-1

Given equation is \$3"x"^2 − 14"x" + 8 = 0\$

To solve this equation, we need to factorize the quadratic equation: case - 1

⇒ \$3"x"^2 − 12"x" − 2"x" + 8 = 0\$

⇒ 3x(x − 4) − 2(x − 4)=0

⇒ (3x − 2)(x − 4) = 0

Explanation: Here, the given quadratic equation is factorized into simple from is (3x - 2) (x - 4) = 0.

Step 1b

Setting each factor equal to zero 3x − 2 = 0 and x − 4 = 0.

Explanation: Equating each factor to zero, we distinctively isolate them.

Step 1c

Solving these equations gives us 3x = 2 and x = 4.

\$"x" = 2/3\$ and x = 4

Explanation: Here, the slove for x values are \$"x" = 2/3\$ and x = 4.

Step 2a

Case-2

Given quadratic equation is \$3"x"^2 − 14"x" + 8 = 0\$

Identify the coefficients a, b, and c in the quadratic equation: Case - 2

\$"ax"^2 + "bx" + "c" = "0" \$

a = 3 , b = -14 , c = 8

Explanation: Here, the values are given to the coefficients.

Step 2b

Use the quadratic formula: \$ "x" = (-b ± \sqrt (b^2 - 4ac))/(2a) \$

\$"x" = ((-(-14) ± \sqrt ((-14)^2 - 4(3)(8)))/((2)(3))) \$

\$"x" = (14 ± \sqrt (100)) / (6) \$

Explanation: Here, substitute the values of a, b, and c into the quadratic formula.

Step 2c

Simplify the expression under the square root:

\$ "x" = ((14 ± 10) / (6)) \$

x = 4 and x = \$ 2/3 \$

Explanation: Here, slove for x values are \$"x" = 2/3\$ and x = 4.