Algebra Examples

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

Step 1

Add $3$ to both sides of the equation.

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

Step 2

Divide each term in $2 x^{2} + 5 x = 3$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x^{2} + 5 x = 3$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{5 x}{2} = \frac{3}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{5 x}{2} = \frac{3}{2}$

Step 2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{5 x}{2} = \frac{3}{2}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{5}{4})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{5 x}{2} + {(\frac{5}{4})}^{2} = \frac{3}{2} + {(\frac{5}{4})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $\frac{5}{4}$ .

$x^{2} + \frac{5 x}{2} + \frac{5^{2}}{4^{2}} = \frac{3}{2} + {(\frac{5}{4})}^{2}$

Step 5.1.1.2

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

$x^{2} + \frac{5 x}{2} + \frac{25}{4^{2}} = \frac{3}{2} + {(\frac{5}{4})}^{2}$

Step 5.1.1.3

Raise $4$ to the power of $2$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3}{2} + {(\frac{5}{4})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{3}{2} + {(\frac{5}{4})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the product rule to $\frac{5}{4}$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3}{2} + \frac{5^{2}}{4^{2}}$

Step 5.2.1.1.2

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

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3}{2} + \frac{25}{4^{2}}$

Step 5.2.1.1.3

Raise $4$ to the power of $2$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3}{2} + \frac{25}{16}$

Step 5.2.1.2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3}{2} \cdot \frac{8}{8} + \frac{25}{16}$

Step 5.2.1.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{3}{2}$ by $\frac{8}{8}$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3 \cdot 8}{2 \cdot 8} + \frac{25}{16}$

Step 5.2.1.3.2

Multiply $2$ by $8$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3 \cdot 8}{16} + \frac{25}{16}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{3 \cdot 8 + 25}{16}$

Step 5.2.1.5

Simplify the numerator.

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Step 5.2.1.5.1

Multiply $3$ by $8$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{24 + 25}{16}$

Step 5.2.1.5.2

Add $24$ and $25$ .

$x^{2} + \frac{5 x}{2} + \frac{25}{16} = \frac{49}{16}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{5}{4})}^{2}$ .

${(x + \frac{5}{4})}^{2} = \frac{49}{16}$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{5}{4} = \pm \sqrt{\frac{49}{16}}$

Step 7.2

Simplify $\pm \sqrt{\frac{49}{16}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{49}{16}}$ as $\frac{\sqrt{49}}{\sqrt{16}}$ .

$x + \frac{5}{4} = \pm \frac{\sqrt{49}}{\sqrt{16}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $49$ as $7^{2}$ .

$x + \frac{5}{4} = \pm \frac{\sqrt{7^{2}}}{\sqrt{16}}$

Step 7.2.2.2

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

$x + \frac{5}{4} = \pm \frac{7}{\sqrt{16}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $16$ as $4^{2}$ .

$x + \frac{5}{4} = \pm \frac{7}{\sqrt{4^{2}}}$

Step 7.2.3.2

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

$x + \frac{5}{4} = \pm \frac{7}{4}$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x + \frac{5}{4} = \frac{7}{4}$

Step 7.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.2.1

Subtract $\frac{5}{4}$ from both sides of the equation.

$x = \frac{7}{4} - \frac{5}{4}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$x = \frac{7 - 5}{4}$

Step 7.3.2.3

Subtract $5$ from $7$ .

$x = \frac{2}{4}$

Step 7.3.2.4

Cancel the common factor of $2$ and $4$ .

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Step 7.3.2.4.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{4}$

Step 7.3.2.4.2

Cancel the common factors.

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Step 7.3.2.4.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 1}{2 \cdot 2}$

Step 7.3.2.4.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 2}$

Step 7.3.2.4.2.3

Rewrite the expression.

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

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + \frac{5}{4} = - \frac{7}{4}$

Step 7.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.4.1

Subtract $\frac{5}{4}$ from both sides of the equation.

$x = - \frac{7}{4} - \frac{5}{4}$

Step 7.3.4.2

Combine the numerators over the common denominator.

$x = \frac{- 7 - 5}{4}$

Step 7.3.4.3

Subtract $5$ from $- 7$ .

$x = \frac{- 12}{4}$

Step 7.3.4.4

Divide $- 12$ by $4$ .

$x = - 3$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{2}, - 3$