Algebra Examples

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

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

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

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Step 1.1.1

Subtract $5 x^{2}$ from both sides of the equation.

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

Step 1.1.2

Add $6$ to both sides of the equation.

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

Step 1.2

Subtract $5 x^{2}$ from $2 x^{2}$ .

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

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 3 \cdot 6}}{2 \cdot - 3}$

Step 4.1.2

Multiply $- 4 \cdot - 3 \cdot 6$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 3 \pm \sqrt{9 + 12 \cdot 6}}{2 \cdot - 3}$

Step 4.1.2.2

Multiply $12$ by $6$ .

$x = \frac{- 3 \pm \sqrt{9 + 72}}{2 \cdot - 3}$

Step 4.1.3

Add $9$ and $72$ .

$x = \frac{- 3 \pm \sqrt{81}}{2 \cdot - 3}$

Step 4.1.4

Rewrite $81$ as $9^{2}$ .

$x = \frac{- 3 \pm \sqrt{9^{2}}}{2 \cdot - 3}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 3 \pm 9}{2 \cdot - 3}$

Step 4.2

Multiply $2$ by $- 3$ .

$x = \frac{- 3 \pm 9}{- 6}$

Step 4.3

Simplify $\frac{- 3 \pm 9}{- 6}$ .

$x = \frac{1 \pm 3}{2}$

Step 5

The final answer is the combination of both solutions.

$x = 2, - 1$