Finite Math Examples

$\frac{x - 2}{\sqrt{4 x^{2} - 19 x + 30}} = \frac{1}{3}$

Step 1

Cross multiply.

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Step 1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$1 \cdot (\sqrt{4 x^{2} - 19 x + 30}) = (x - 2) \cdot (3)$

Step 1.2

Simplify the left side.

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Step 1.2.1

Multiply $\sqrt{4 x^{2} - 19 x + 30}$ by $1$ .

$\sqrt{4 x^{2} - 19 x + 30} = (x - 2) \cdot (3)$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify $(x - 2) \cdot (3)$ .

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Step 1.3.1.1

Apply the distributive property.

$\sqrt{4 x^{2} - 19 x + 30} = x \cdot 3 - 2 \cdot 3$

Step 1.3.1.2

Simplify the expression.

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Step 1.3.1.2.1

Move $3$ to the left of $x$ .

$\sqrt{4 x^{2} - 19 x + 30} = 3 \cdot x - 2 \cdot 3$

Step 1.3.1.2.2

Multiply $- 2$ by $3$ .

$\sqrt{4 x^{2} - 19 x + 30} = 3 x - 6$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 x^{2} - 19 x + 30}}^{2} = {(3 x - 6)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x^{2} - 19 x + 30}$ as ${(4 x^{2} - 19 x + 30)}^{\frac{1}{2}}$ .

${({(4 x^{2} - 19 x + 30)}^{\frac{1}{2}})}^{2} = {(3 x - 6)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(4 x^{2} - 19 x + 30)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(4 x^{2} - 19 x + 30)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(4 x^{2} - 19 x + 30)}^{\frac{1}{2} \cdot 2} = {(3 x - 6)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(4 x^{2} - 19 x + 30)}^{\frac{1}{2} \cdot 2} = {(3 x - 6)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(4 x^{2} - 19 x + 30)}^{1} = {(3 x - 6)}^{2}$

Step 3.2.1.2

Simplify.

$4 x^{2} - 19 x + 30 = {(3 x - 6)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 x - 6)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(3 x - 6)}^{2}$ as $(3 x - 6) (3 x - 6)$ .

$4 x^{2} - 19 x + 30 = (3 x - 6) (3 x - 6)$

Step 3.3.1.2

Expand $(3 x - 6) (3 x - 6)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$4 x^{2} - 19 x + 30 = 3 x (3 x - 6) - 6 (3 x - 6)$

Step 3.3.1.2.2

Apply the distributive property.

$4 x^{2} - 19 x + 30 = 3 x (3 x) + 3 x \cdot - 6 - 6 (3 x - 6)$

Step 3.3.1.2.3

Apply the distributive property.

$4 x^{2} - 19 x + 30 = 3 x (3 x) + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$4 x^{2} - 19 x + 30 = 3 \cdot 3 x \cdot x + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6$

Step 3.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.3.1.2.1

Move $x$ .

$4 x^{2} - 19 x + 30 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6$

Step 3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$4 x^{2} - 19 x + 30 = 3 \cdot 3 x^{2} + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6$

Step 3.3.1.3.1.3

Multiply $3$ by $3$ .

$4 x^{2} - 19 x + 30 = 9 x^{2} + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6$

Step 3.3.1.3.1.4

Multiply $- 6$ by $3$ .

$4 x^{2} - 19 x + 30 = 9 x^{2} - 18 x - 6 (3 x) - 6 \cdot - 6$

Step 3.3.1.3.1.5

Multiply $3$ by $- 6$ .

$4 x^{2} - 19 x + 30 = 9 x^{2} - 18 x - 18 x - 6 \cdot - 6$

Step 3.3.1.3.1.6

Multiply $- 6$ by $- 6$ .

$4 x^{2} - 19 x + 30 = 9 x^{2} - 18 x - 18 x + 36$

Step 3.3.1.3.2

Subtract $18 x$ from $- 18 x$ .

$4 x^{2} - 19 x + 30 = 9 x^{2} - 36 x + 36$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $9 x^{2}$ from both sides of the equation.

$4 x^{2} - 19 x + 30 - 9 x^{2} = - 36 x + 36$

Step 4.1.2

Add $36 x$ to both sides of the equation.

$4 x^{2} - 19 x + 30 - 9 x^{2} + 36 x = 36$

Step 4.1.3

Subtract $9 x^{2}$ from $4 x^{2}$ .

$- 5 x^{2} - 19 x + 30 + 36 x = 36$

Step 4.1.4

Add $- 19 x$ and $36 x$ .

$- 5 x^{2} + 17 x + 30 = 36$

Step 4.2

Subtract $36$ from both sides of the equation.

$- 5 x^{2} + 17 x + 30 - 36 = 0$

Step 4.3

Subtract $36$ from $30$ .

$- 5 x^{2} + 17 x - 6 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $- 1$ out of $- 5 x^{2} + 17 x - 6$ .

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Step 4.4.1.1

Factor $- 1$ out of $- 5 x^{2}$ .

$- (5 x^{2}) + 17 x - 6 = 0$

Step 4.4.1.2

Factor $- 1$ out of $17 x$ .

$- (5 x^{2}) - (- 17 x) - 6 = 0$

Step 4.4.1.3

Rewrite $- 6$ as $- 1 (6)$ .

$- (5 x^{2}) - (- 17 x) - 1 \cdot 6 = 0$

Step 4.4.1.4

Factor $- 1$ out of $- (5 x^{2}) - (- 17 x)$ .

$- (5 x^{2} - 17 x) - 1 \cdot 6 = 0$

Step 4.4.1.5

Factor $- 1$ out of $- (5 x^{2} - 17 x) - 1 (6)$ .

$- (5 x^{2} - 17 x + 6) = 0$

Step 4.4.2

Factor.

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Step 4.4.2.1

Factor by grouping.

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Step 4.4.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 6 = 30$ and whose sum is $b = - 17$ .

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Step 4.4.2.1.1.1

Factor $- 17$ out of $- 17 x$ .

$- (5 x^{2} - 17 x + 6) = 0$

Step 4.4.2.1.1.2

Rewrite $- 17$ as $- 2$ plus $- 15$

$- (5 x^{2} + (- 2 - 15) x + 6) = 0$

Step 4.4.2.1.1.3

Apply the distributive property.

$- (5 x^{2} - 2 x - 15 x + 6) = 0$

Step 4.4.2.1.2

Factor out the greatest common factor from each group.

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Step 4.4.2.1.2.1

Group the first two terms and the last two terms.

$- ((5 x^{2} - 2 x) - 15 x + 6) = 0$

Step 4.4.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (x (5 x - 2) - 3 (5 x - 2)) = 0$

Step 4.4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x - 2$ .

$- ((5 x - 2) (x - 3)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$- (5 x - 2) (x - 3) = 0$

Step 4.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5 x - 2 = 0$

$x - 3 = 0$

Step 4.6

Set $5 x - 2$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Step 4.6.2

Solve $5 x - 2 = 0$ for $x$ .

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Step 4.6.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 4.6.2.2

Divide each term in $5 x = 2$ by $5$ and simplify.

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Step 4.6.2.2.1

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Step 4.6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 4.7

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.7.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.8

The final solution is all the values that make $- (5 x - 2) (x - 3) = 0$ true.

$x = \frac{2}{5}, 3$

Step 5

Exclude the solutions that do not make $\frac{x - 2}{\sqrt{4 x^{2} - 19 x + 30}} = \frac{1}{3}$ true.

$x = 3$