Given,
Equation,
3x² - 2x - 3 = 0
To find,
Solution using the quadratic formula.
Solution,
Using the quadratic formula, the solution to a given quadratic equation or roots of the equation can be found. Suppose the quadratic equation is,
ax² + bx + c = 0,
Then its solution or roots will be given by the formula,
[tex]x_{1,2}=\frac{-b\pm \sqrt{b^{2} -4ac} }{2a}[/tex]
Here, the given equation is,
3x² - 2x - 3 = 0
So,
a = 3,
b = -2, and,
c = -3.
Substituting these values in the above formula,
[tex]x_{1,2}=\frac{-(-2)\pm \sqrt{(-2)^{2} -4(3)(-3)} }{2(3)}[/tex]
⇒ [tex]x_{1,2}=\frac{2\pm \sqrt{(4) +36} }{6}[/tex]
⇒ [tex]x_{1,2}=\frac{2\pm \sqrt{40} }{6}[/tex]
⇒ [tex]x_{1,2}=\frac{2\pm 2\sqrt{10} }{6}[/tex]
⇒ [tex]x_{1,2}=\frac{1\pm \sqrt{10} }{3}[/tex]
⇒ [tex]x_{1}=\frac{1 + \sqrt{10} }{3},x_{2}=\frac{1 - \sqrt{10} }{3},[/tex]
Therefore, the solution of the given equations using quadratic formula will be [tex]x_{1}=\frac{1 + \sqrt{10} }{3},x_{2}=\frac{1 - \sqrt{10} }{3}.[/tex]