Precalculus Examples

$5 x = 3 x^{2} + 1$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $3 x^{2}$ from both sides of the equation.

$5 x - 3 x^{2} = 1$

Step 1.2

Subtract $1$ from both sides of the equation.

$5 x - 3 x^{2} - 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 3$ , $b = 5$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 3 \cdot - 1)}}{2 \cdot - 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 3 \cdot - 1}}{2 \cdot - 3}$

Step 4.1.2

Multiply $- 4 \cdot - 3 \cdot - 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 5 \pm \sqrt{25 + 12 \cdot - 1}}{2 \cdot - 3}$

Step 4.1.2.2

Multiply $12$ by $- 1$ .

$x = \frac{- 5 \pm \sqrt{25 - 12}}{2 \cdot - 3}$

Step 4.1.3

Subtract $12$ from $25$ .

$x = \frac{- 5 \pm \sqrt{13}}{2 \cdot - 3}$

Step 4.2

Multiply $2$ by $- 3$ .

$x = \frac{- 5 \pm \sqrt{13}}{- 6}$

Step 4.3

Simplify $\frac{- 5 \pm \sqrt{13}}{- 6}$ .

$x = \frac{5 \pm \sqrt{13}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{13}}{6}$

Decimal Form:

$x = 1.43425854 \dots, 0.23240812 \dots$