Solve for x 5x=6x^2-3

$5 x = 6 x^{2} - 3$

Subtract $6 x^{2}$ from both sides of the equation.

$5 x - 6 x^{2} = - 3$

Add $3$ to both sides of the equation.

$5 x - 6 x^{2} + 3 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 6$ , $b = 5$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 6 \cdot 3)}}{2 \cdot - 6}$

Simplify.

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Simplify the numerator.

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Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 6 \cdot 3}}{2 \cdot - 6}$

Multiply $- 4$ by $- 6$ .

$x = \frac{- 5 \pm \sqrt{25 + 24 \cdot 3}}{2 \cdot - 6}$

Multiply $24$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 + 72}}{2 \cdot - 6}$

Add $25$ and $72$ .

$x = \frac{- 5 \pm \sqrt{97}}{2 \cdot - 6}$

Multiply $2$ by $- 6$ .

$x = \frac{- 5 \pm \sqrt{97}}{- 12}$

Simplify $\frac{- 5 \pm \sqrt{97}}{- 12}$ .

$x = \frac{5 \pm \sqrt{97}}{12}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{97}}{12}, \frac{5 - \sqrt{97}}{12}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{97}}{12}, \frac{5 - \sqrt{97}}{12}$

Decimal Form:

$x = 1.23740481 \dots, - 0.40407148 \dots$