Question

solve using the quadratic FORMULAshow all work3x^2=5x+2

Answer 1

The given quadratic equation is

$3x^2=5x+2$

First of all, let us re-write the equation in standard form.

$3x^2-5x-2=0$

Recall that the standard form of a quadratic equation is given by

$ax^2+bx+c=0$

Comparing the standard form with the given equation we see that,

a = 3

b = -5

c = -2

Now we can solve using the quadratic formula.

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

Let us substitute the values of the coefficients a, b, c

$\begin{gathered} x=\frac{-(-5)\pm\sqrt[]{(-5)^2-4(3)(-2)}}{2(3)} \\ x=\frac{5\pm\sqrt[]{25^{}-(-24)}}{6}=\frac{5\pm\sqrt[]{25^{}+24}}{6}=\frac{5\pm\sqrt[]{49}}{6}=\frac{5\pm7}{6} \end{gathered}$

So, the two possible solutions of the quadratic equation are

$\begin{gathered} x_1=\frac{5+7}{6}\: \text{and}\: x_2=\frac{5-7}{6} \\ x_1=\frac{12}{6}\: \text{and}\: x_2=\frac{-2}{6} \\ x_1=2\: \text{and}\: x_2=-\frac{1}{3} \end{gathered}$

Therefore, the solution of the given quadratic equation is

$x=(2,-\frac{1}{3})$