Algebra Examples

$2 x^{2} + 5 x - 8 = \frac{5}{2} x + 20$

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

Combine $\frac{5}{2}$ and $x$ .

$2 x^{2} + 5 x - 8 = \frac{5 x}{2} + 20$

Step 1.2

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

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Step 1.2.1

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

$2 x^{2} + 5 x - 8 - \frac{5 x}{2} = 20$

Step 1.2.2

Subtract $20$ from both sides of the equation.

$2 x^{2} + 5 x - 8 - \frac{5 x}{2} - 20 = 0$

Step 1.3

Subtract $20$ from $- 8$ .

$2 x^{2} + 5 x - \frac{5 x}{2} - 28 = 0$

Step 2

Multiply through by the least common denominator $2$ , then simplify.

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Step 2.1

To write $5 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 (2 x^{2} + 5 x \cdot \frac{2}{2} - \frac{5 x}{2} - 28) = 0$

Step 2.2

Combine $5 x$ and $\frac{2}{2}$ .

$2 (2 x^{2} + \frac{5 x \cdot 2}{2} - \frac{5 x}{2} - 28) = 0$

Step 2.3

Combine the numerators over the common denominator.

$2 (2 x^{2} + \frac{5 x \cdot 2 - 5 x}{2} - 28) = 0$

Step 2.4

Find the common denominator.

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Step 2.4.1

Write $2 x^{2}$ as a fraction with denominator $1$ .

$2 (\frac{2 x^{2}}{1} + \frac{5 x \cdot 2 - 5 x}{2} - 28) = 0$

Step 2.4.2

Multiply $\frac{2 x^{2}}{1}$ by $\frac{2}{2}$ .

$2 (\frac{2 x^{2}}{1} \cdot \frac{2}{2} + \frac{5 x \cdot 2 - 5 x}{2} - 28) = 0$

Step 2.4.3

Multiply $\frac{2 x^{2}}{1}$ by $\frac{2}{2}$ .

$2 (\frac{2 x^{2} \cdot 2}{2} + \frac{5 x \cdot 2 - 5 x}{2} - 28) = 0$

Step 2.4.4

Write $- 28$ as a fraction with denominator $1$ .

$2 (\frac{2 x^{2} \cdot 2}{2} + \frac{5 x \cdot 2 - 5 x}{2} + \frac{- 28}{1}) = 0$

Step 2.4.5

Multiply $\frac{- 28}{1}$ by $\frac{2}{2}$ .

$2 (\frac{2 x^{2} \cdot 2}{2} + \frac{5 x \cdot 2 - 5 x}{2} + \frac{- 28}{1} \cdot \frac{2}{2}) = 0$

Step 2.4.6

Multiply $\frac{- 28}{1}$ by $\frac{2}{2}$ .

$2 (\frac{2 x^{2} \cdot 2}{2} + \frac{5 x \cdot 2 - 5 x}{2} + \frac{- 28 \cdot 2}{2}) = 0$

Step 2.5

Combine the numerators over the common denominator.

$2 (\frac{2 x^{2} \cdot 2 + 5 x \cdot 2 - 5 x - 28 \cdot 2}{2}) = 0$

Step 2.6

Simplify each term.

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Step 2.6.1

Multiply $2$ by $2$ .

$2 (\frac{4 x^{2} + 5 x \cdot 2 - 5 x - 28 \cdot 2}{2}) = 0$

Step 2.6.2

Multiply $2$ by $5$ .

$2 (\frac{4 x^{2} + 10 x - 5 x - 28 \cdot 2}{2}) = 0$

Step 2.6.3

Multiply $- 28$ by $2$ .

$2 (\frac{4 x^{2} + 10 x - 5 x - 56}{2}) = 0$

Step 2.7

Subtract $5 x$ from $10 x$ .

$2 (\frac{4 x^{2} + 5 x - 56}{2}) = 0$

Step 2.8

Cancel the common factor of $2$ .

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Step 2.8.1

Cancel the common factor.

$2 (\frac{4 x^{2} + 5 x - 56}{2}) = 0$

Step 2.8.2

Rewrite the expression.

$4 x^{2} + 5 x - 56 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = 5$ , and $c = - 56$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (4 \cdot - 56)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 4 \cdot - 56}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 56$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 5 \pm \sqrt{25 - 16 \cdot - 56}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 56$ .

$x = \frac{- 5 \pm \sqrt{25 + 896}}{2 \cdot 4}$

Step 5.1.3

Add $25$ and $896$ .

$x = \frac{- 5 \pm \sqrt{921}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 5 \pm \sqrt{921}}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 5 \pm \sqrt{921}}{8}$

Decimal Form:

$x = 3.16849772 \dots, - 4.41849772 \dots$