Algebra Examples

$2 x^{2} = 10 x + 3$

Step 1

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

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Step 1.1

Subtract $10 x$ from both sides of the equation.

$2 x^{2} - 10 x = 3$

Step 1.2

Subtract $3$ from both sides of the equation.

$2 x^{2} - 10 x - 3 = 0$

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 $a = 2$ , $b = - 10$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (2 \cdot - 3)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot - 3}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot - 3$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{10 \pm \sqrt{100 - 8 \cdot - 3}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $- 3$ .

$x = \frac{10 \pm \sqrt{100 + 24}}{2 \cdot 2}$

Step 4.1.3

Add $100$ and $24$ .

$x = \frac{10 \pm \sqrt{124}}{2 \cdot 2}$

Step 4.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 4.1.4.1

Factor $4$ out of $124$ .

$x = \frac{10 \pm \sqrt{4 (31)}}{2 \cdot 2}$

Step 4.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 2}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{31}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{10 \pm 2 \sqrt{31}}{4}$

Step 4.3

Simplify $\frac{10 \pm 2 \sqrt{31}}{4}$ .

$x = \frac{5 \pm \sqrt{31}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{31}}{2}$

Decimal Form:

$x = 5.28388218 \dots, - 0.28388218 \dots$