Finite Math Examples

$3 x \cdot 1 \cdot (2 x \cdot 2) = 9$ , $6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Multiply $3$ by $1$ .

$3 x \cdot (2 x \cdot 2) = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 1.2

Multiply $2$ by $2$ .

$3 x \cdot (4 x) = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 1.3

Multiply $4$ by $3$ .

$12 x \cdot x = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 1.4

Raise $x$ to the power of $1$ .

$12 (x \cdot x) = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 1.5

Raise $x$ to the power of $1$ .

$12 (x \cdot x) = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{1 + 1} = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 1.7

Add $1$ and $1$ .

$12 x^{2} = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

$12 x^{2} = 9$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 2

Divide each term in $12 x^{2} = 9$ by $12$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $12 x^{2} = 9$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{9}{12}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{9}{12}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 2.2.1.2

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

$x^{2} = \frac{9}{12}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

$x^{2} = \frac{9}{12}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

$x^{2} = \frac{9}{12}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Cancel the common factor of $9$ and $12$ .

Tap for more steps...

Step 2.3.1.1

Factor $3$ out of $9$ .

$x^{2} = \frac{3 (3)}{12}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.2.1

Factor $3$ out of $12$ .

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 2.3.1.2.2

Cancel the common factor.

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 2.3.1.2.3

Rewrite the expression.

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 3

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

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 4

Simplify $\pm \sqrt{\frac{3}{4}}$ .

Tap for more steps...

Step 4.1

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 4.2

Simplify the denominator.

Tap for more steps...

Step 4.2.1

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

$x = \pm \frac{\sqrt{3}}{\sqrt{2^{2}}}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 4.2.2

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

$x = \pm \frac{\sqrt{3}}{2}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

$x = \pm \frac{\sqrt{3}}{2}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

$x = \pm \frac{\sqrt{3}}{2}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 5

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

Tap for more steps...

Step 5.1

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

$x = \frac{\sqrt{3}}{2}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 5.2

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

$x = - \frac{\sqrt{3}}{2}$

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 5.3

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

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

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

$6 x \cdot 1 - 5 x \cdot 2 = 9$

Step 6

Simplify $6 x \cdot 1 - 5 x \cdot 2$ .

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Multiply $6$ by $1$ .

$6 x - 5 x \cdot 2 = 9$

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

Step 6.1.2

Multiply $2$ by $- 5$ .

$6 x - 10 x = 9$

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

$6 x - 10 x = 9$

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

Step 6.2

Subtract $10 x$ from $6 x$ .

$- 4 x = 9$

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

$- 4 x = 9$

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

Step 7

Divide each term in $- 4 x = 9$ by $- 4$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $- 4 x = 9$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{9}{- 4}$

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

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{9}{- 4}$

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

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{- 4}$

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

$x = \frac{9}{- 4}$

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

$x = \frac{9}{- 4}$

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

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Move the negative in front of the fraction.

$x = - \frac{9}{4}$

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

$x = - \frac{9}{4}$

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

$x = - \frac{9}{4}$

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

Step 8

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(x - (\frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2})) (x - (- \frac{9}{4})) = 0$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify terms.

Tap for more steps...

Step 9.1.1

Apply the distributive property.

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + (x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) (\frac{9}{4}) = 0$

Step 9.1.2

Combine $(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2})$ and $\frac{9}{4}$ .

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) \cdot 9}{4} = 0$

Step 9.2

Simplify each term.

Tap for more steps...

Step 9.2.1

Simplify the numerator.

Tap for more steps...

Step 9.2.1.1

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

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(x \cdot \frac{2}{2} - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) \cdot 9}{4} = 0$

Step 9.2.1.2

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

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(\frac{x \cdot 2}{2} - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) \cdot 9}{4} = 0$

Step 9.2.1.3

Combine the numerators over the common denominator.

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(\frac{x \cdot 2 - \sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) \cdot 9}{4} = 0$

Step 9.2.1.4

Move $2$ to the left of $x$ .

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(\frac{2 x - \sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) \cdot 9}{4} = 0$

Step 9.2.2

Move $9$ to the left of $(\frac{2 x - \sqrt{3}}{2}, - \frac{\sqrt{3}}{2})$ .

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{9 \cdot (\frac{2 x - \sqrt{3}}{2}, - \frac{\sqrt{3}}{2})}{4} = 0$

Step 9.2.3

Simplify the numerator.

Tap for more steps...

Step 9.2.3.1

Multiply $9$ by each element of the matrix.

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(9 (\frac{2 x - \sqrt{3}}{2}), 9 (- \frac{\sqrt{3}}{2}))}{4} = 0$

Step 9.2.3.2

Simplify each element in the matrix.

Tap for more steps...

Step 9.2.3.2.1

Combine $9$ and $\frac{2 x - \sqrt{3}}{2}$ .

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(\frac{9 (2 x - \sqrt{3})}{2}, 9 (- \frac{\sqrt{3}}{2}))}{4} = 0$

Step 9.2.3.2.2

Multiply $9 (- \frac{\sqrt{3}}{2})$ .

Tap for more steps...

Step 9.2.3.2.2.1

Multiply $- 1$ by $9$ .

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(\frac{9 (2 x - \sqrt{3})}{2}, - 9 \frac{\sqrt{3}}{2})}{4} = 0$

Step 9.2.3.2.2.2

Combine $- 9$ and $\frac{\sqrt{3}}{2}$ .

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(\frac{9 (2 x - \sqrt{3})}{2}, \frac{- 9 \sqrt{3}}{2})}{4} = 0$

Step 9.2.3.2.3

Move the negative in front of the fraction.

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x + \frac{(\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.3

To write $(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x \cdot \frac{4}{4} + \frac{(\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.4

Simplify terms.

Tap for more steps...

Step 9.4.1

Combine $(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x$ and $\frac{4}{4}$ .

$\frac{(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x \cdot 4}{4} + \frac{(\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.4.2

Combine the numerators over the common denominator.

$\frac{(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x \cdot 4 + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5

Simplify the numerator.

Tap for more steps...

Step 9.5.1

Move $4$ to the left of $(x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x$ .

$\frac{4 \cdot ((x - \frac{\sqrt{3}}{2}, - \frac{\sqrt{3}}{2}) x) + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.2

Multiply $4$ by each element of the matrix.

$\frac{(4 (x - \frac{\sqrt{3}}{2}), 4 (- \frac{\sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3

Simplify each element in the matrix.

Tap for more steps...

Step 9.5.3.1

Apply the distributive property.

$\frac{(4 x + 4 (- \frac{\sqrt{3}}{2}), 4 (- \frac{\sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.5.3.2.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$\frac{(4 x + 4 (\frac{- \sqrt{3}}{2}), 4 (- \frac{\sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.2.2

Factor $2$ out of $4$ .

$\frac{(4 x + 2 (2) (\frac{- \sqrt{3}}{2}), 4 (- \frac{\sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.2.3

Cancel the common factor.

$\frac{(4 x + 2 \cdot (2 (\frac{- \sqrt{3}}{2})), 4 (- \frac{\sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.2.4

Rewrite the expression.

$\frac{(4 x + 2 (- \sqrt{3}), 4 (- \frac{\sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.3

Multiply $- 1$ by $2$ .

$\frac{(4 x - 2 \sqrt{3}, 4 (- \frac{\sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.5.3.4.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$\frac{(4 x - 2 \sqrt{3}, 4 (\frac{- \sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.4.2

Factor $2$ out of $4$ .

$\frac{(4 x - 2 \sqrt{3}, 2 (2) (\frac{- \sqrt{3}}{2})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.4.3

Cancel the common factor.

$\frac{(4 x - 2 \sqrt{3}, 2 \cdot (2 (\frac{- \sqrt{3}}{2}))) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.4.4

Rewrite the expression.

$\frac{(4 x - 2 \sqrt{3}, 2 (- \sqrt{3})) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$

Step 9.5.3.5

Multiply $- 1$ by $2$ .

$\frac{(4 x - 2 \sqrt{3}, - 2 \sqrt{3}) x + (\frac{9 (2 x - \sqrt{3})}{2}, - \frac{9 \sqrt{3}}{2})}{4} = 0$