Question

Solve the quadratic equation 3x^2-5x-7=0

Answer 1

Explanation:

`3x^(2) -5x -7 = 0 \\ 3x^(2) -5x = 7 \\ x^(2) -(5)/(3)x = (7)/(3) \\ x^(2) -(5)/(3)x +(25)/(36) = (7)/(3) +(25)/(36) \\ (x -(5)/(6))^(2) = (84 +25)/(36) \\ (x -(5)/(6))^(2) = (109)/(36) \\ x -(5)/(6) = ±\sqrt{(109)/(36)} \\ x -(5)/(6) = ±(√(109))/(6) \\ x = ±(√(109))/(6) +(5)/(6)`

Equation from the positive root:

`x = (√(109))/(6) +(5)/(6) \\ x = (√(109) +5)/(6)`

Equation from the negative root:

`x = -(√(109))/(6) +(5)/(6) \\ x = (-√(109) +5)/(6)`

Answer:

`x = (√(109) +5)/(6)\\` and
`x = (-√(109) +5)/(6)\\` satisfy your quadratic equation.

Answer 2

Given : A quadratic equation is given to us . The equation is 3x² - 5x - 7 = 0 .

To Find : The roots of the equation .

Solution : Given quadratic equation is 3x²-5x-7=0. So , let's factorise it to get the zeroes of the equation .

⇒ 3x² -5x - 7 = 0.

Here now , use Quadratic formula , of the quadratic equation in standard form of ax² + bx + c = 0.

`\boxed{\red{\bf x\:\:=\:\:(-b\pm√(b^2-4ac))/(2a)}}`

On substituting respective values ,

⇒ x = -(-5) ± √ (-5)² - 4×3×(-7) / 2 × 3.

⇒x = 5 ± √ [ 25 + 84 ]/ 6 .

⇒ x = 5 ± √ 109 / 6 .

⇒ x = 5 + √109 / 6 , 5 - √109 / 6.

`\purple{\underbrace{\underline{\boxed{\red{\tt \purple{\longmapsto} x = (5+√(109))/(6), (5-√(109))/(6)}}}}}`