Question

Use the quadratic formula to solve for the roots in the following equation. 4x 2 + 5x + 2 = 2x 2 + 7x – 1

Answer 1

Answer:

So, The roots are $x= \frac{1+\sqrt{5}i}{2} \,\, and \,\,x= \frac{1-\sqrt{5}i}{2}$

Step-by-step explanation:

$4x^2 + 5x + 2 = 2x^2 + 7x - 1$

We need to solve the equation to find the roots using quadratic formula.

The quadratic formula is:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Rearranging the above equation:

$4x^2 -2x^2+ 5x-7x + 2+1 =0$

$2x^2 -2x + 3 =0$

Where a =2 , b=-2 and c =3 Putting values in quadratic equation and solving:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\x=\frac{-(-2)\pm\sqrt{(-2)^2-4(2)(3)}}{2(2)}\\x=\frac{2\pm\sqrt{4-24}}{4}\\x=\frac{2\pm\sqrt{-20}}{4}\\\sqrt{-20} \,\,can\,\,be\,\, written\,\, as\,\, 2\sqrt{-5}\\ x=\frac{2\pm2\sqrt{-5}}{4}\\x=\frac{2+2\sqrt{-5}}{4} \,\, and \,\, x=\frac{2-2\sqrt{-5}}{4}\\x=\frac{2(1+\sqrt{-5})}{4} \,\, and \,\, x=\frac{2(1-\sqrt{-5})}{4}\\ x=\frac{1+\sqrt{-5}}{2} \,\, and \,\, x=\frac{1-\sqrt{-5}}{2}\\As \,\,we\,\, know\,\, \sqrt{-1} = i$

$x= \frac{1+\sqrt{5}i}{2} \,\, and \,\,x= \frac{1-\sqrt{5}i}{2}$

So, The roots are $x= \frac{1+\sqrt{5}i}{2} \,\, and \,\,x= \frac{1-\sqrt{5}i}{2}$