Question

How do you solve $ 3{x^2} + 4x = 2 $ ?

Answer 1

Hint: The highest exponent of the polynomial is known as the degree of the polynomial equation, when its degree is two then the polynomial equation is called a quadratic equation. The values of the unknown variable for which the value of the function comes out to be zero are called the zeros/factors/solutions of the polynomial equation. Factorization, completing the square, graphs, quadratic formula, etc. are some methods for finding the roots of a quadratic equation. We have to solve the quadratic equation given in the question so we will first find out the best method for finding the solutions, we will rewrite the equation in its standard form and then solve it.

Complete step-by-step answer:
The equation given is $ 3{x^2} + 4x = 2 $
It can be rewritten as - $ 3{x^2} + 4x - 2 = 0 $
The above equation cannot be solved by factorization, so we have to use the quadratic formula.
On comparing the given equation with the standard quadratic equation $ a{x^2} + bx + c = 0 $ , we get –
 $ a = 3,\,b = 4,\,c = - 2 $
The Quadratic formula is given as –
 $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $
Putting the values in the above equation, we get –
 $
  x = \dfrac{{ - 4 \pm \sqrt {{{(4)}^2} - 4(3)( - 2)} }}{{2(3)}} \\
  x = \dfrac{{ - 4 \pm \sqrt {16 + 24} }}{6} \\
  x = \dfrac{{ - 4 \pm \sqrt {40} }}{6} \\
   \Rightarrow x = \dfrac{{ - 4 \pm 2\sqrt {10} }}{6} \\
   \Rightarrow x = \dfrac{{ - 2 \pm \sqrt {10} }}{3} \;
  $
Hence the zeros of the given equation are $ \dfrac{{ - 2 + \sqrt {10} }}{3} $ and $ \dfrac{{ - 2 - \sqrt {10} }}{3} $ .
So, the correct answer is “ $ \dfrac{{ - 2 + \sqrt {10} }}{3} $ and $ \dfrac{{ - 2 - \sqrt {10} }}{3} $ ”.

Note: The condition for an equation to be a polynomial equation is that the exponent should be a non-negative integer. The value of y is zero on the x-axis, so the roots of an equation are simply the x-intercepts. We are not able to write -6 as a product of two integers such that their sum is equal to 4, so the factors cannot be made that’s why we used the quadratic formula