Pre-Algebra Examples

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

Step 1

Subtract $5 x$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Simplify.

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Step 2.2.1

Multiply $4$ by $2$ .

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

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

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

Step 2.2.3

Multiply $- 5$ by $2$ .

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

Step 2.3

Move $3$ .

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

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

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $8$ .

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

Step 6.1.2.2

Multiply $- 32$ by $3$ .

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

Step 6.1.3

Subtract $96$ from $100$ .

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

Step 6.1.4

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

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

Step 6.1.5

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

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

Step 6.2

Multiply $2$ by $8$ .

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

Step 6.3

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

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

Step 6.4

Change the $\pm$ to $+$ .

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

Step 6.5

Add $5$ and $1$ .

$x = \frac{6}{8}$

Step 6.6

Cancel the common factor of $6$ and $8$ .

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Step 6.6.1

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{8}$

Step 6.6.2

Cancel the common factors.

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Step 6.6.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 3}{2 \cdot 4}$

Step 6.6.2.2

Cancel the common factor.

$x = \frac{2 \cdot 3}{2 \cdot 4}$

Step 6.6.2.3

Rewrite the expression.

$x = \frac{3}{4}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $8$ .

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

Step 7.1.2.2

Multiply $- 32$ by $3$ .

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

Step 7.1.3

Subtract $96$ from $100$ .

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

Step 7.1.4

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

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

Step 7.1.5

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

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

Step 7.2

Multiply $2$ by $8$ .

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

Step 7.3

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

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

Step 7.4

Change the $\pm$ to $-$ .

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

Step 7.5

Subtract $1$ from $5$ .

$x = \frac{4}{8}$

Step 7.6

Cancel the common factor of $4$ and $8$ .

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Step 7.6.1

Factor $4$ out of $4$ .

$x = \frac{4 (1)}{8}$

Step 7.6.2

Cancel the common factors.

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Step 7.6.2.1

Factor $4$ out of $8$ .

$x = \frac{4 \cdot 1}{4 \cdot 2}$

Step 7.6.2.2

Cancel the common factor.

$x = \frac{4 \cdot 1}{4 \cdot 2}$

Step 7.6.2.3

Rewrite the expression.

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

Step 8

The final answer is the combination of both solutions.

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