Pre-Algebra Examples

$x^{2} - 1 \frac{1}{4} \cdot x + \frac{3}{8} = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $x^{2} - 1 \frac{1}{4} \cdot x + \frac{3}{8}$ .

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Step 1.1.1

Convert $1 \frac{1}{4}$ to an improper fraction.

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Step 1.1.1.1

A mixed number is an addition of its whole and fractional parts.

$x^{2} - (1 + \frac{1}{4}) \cdot x + \frac{3}{8} = 0$

Step 1.1.1.2

Add $1$ and $\frac{1}{4}$ .

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Step 1.1.1.2.1

Write $1$ as a fraction with a common denominator.

$x^{2} - (\frac{4}{4} + \frac{1}{4}) \cdot x + \frac{3}{8} = 0$

Step 1.1.1.2.2

Combine the numerators over the common denominator.

$x^{2} - \frac{4 + 1}{4} \cdot x + \frac{3}{8} = 0$

Step 1.1.1.2.3

Add $4$ and $1$ .

$x^{2} - \frac{5}{4} \cdot x + \frac{3}{8} = 0$

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Combine $x$ and $\frac{5}{4}$ .

$x^{2} - \frac{x \cdot 5}{4} + \frac{3}{8} = 0$

Step 1.1.2.2

Move $5$ to the left of $x$ .

$x^{2} - \frac{5 x}{4} + \frac{3}{8} = 0$

Step 2

Multiply through by the least common denominator $8$ , then simplify.

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Step 2.1

Apply the distributive property.

$8 x^{2} + 8 (- \frac{5 x}{4}) + 8 (\frac{3}{8}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{5 x}{4}$ into the numerator.

$8 x^{2} + 8 (\frac{- 5 x}{4}) + 8 (\frac{3}{8}) = 0$

Step 2.2.1.2

Factor $4$ out of $8$ .

$8 x^{2} + 4 (2) (\frac{- 5 x}{4}) + 8 (\frac{3}{8}) = 0$

Step 2.2.1.3

Cancel the common factor.

$8 x^{2} + 4 \cdot (2 (\frac{- 5 x}{4})) + 8 (\frac{3}{8}) = 0$

Step 2.2.1.4

Rewrite the expression.

$8 x^{2} + 2 (- 5 x) + 8 (\frac{3}{8}) = 0$

Step 2.2.2

Multiply $- 5$ by $2$ .

$8 x^{2} - 10 x + 8 (\frac{3}{8}) = 0$

Step 2.2.3

Cancel the common factor of $8$ .

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Step 2.2.3.1

Cancel the common factor.

$8 x^{2} - 10 x + 8 (\frac{3}{8}) = 0$

Step 2.2.3.2

Rewrite the expression.

$8 x^{2} - 10 x + 3 = 0$

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 $a = 8$ , $b = - 10$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (8 \cdot 3)}}{2 \cdot 8}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 8 \cdot 3}}{2 \cdot 8}$

Step 5.1.2

Multiply $- 4 \cdot 8 \cdot 3$ .

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Step 5.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{10 \pm \sqrt{100 - 32 \cdot 3}}{2 \cdot 8}$

Step 5.1.2.2

Multiply $- 32$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 96}}{2 \cdot 8}$

Step 5.1.3

Subtract $96$ from $100$ .

$x = \frac{10 \pm \sqrt{4}}{2 \cdot 8}$

Step 5.1.4

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

$x = \frac{10 \pm \sqrt{2^{2}}}{2 \cdot 8}$

Step 5.1.5

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

$x = \frac{10 \pm 2}{2 \cdot 8}$

Step 5.2

Multiply $2$ by $8$ .

$x = \frac{10 \pm 2}{16}$

Step 5.3

Simplify $\frac{10 \pm 2}{16}$ .

$x = \frac{5 \pm 1}{8}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{3}{4}, \frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{4}, \frac{1}{2}$

Decimal Form:

$x = 0.75, 0.5$