Finite Math Examples

$4 (5 \sqrt{x} - 9) - 7 (\sqrt{x} - 3) = 3 \sqrt{x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 \sqrt{x} = 4 (5 \sqrt{x} - 9) - 7 (\sqrt{x} - 3)$

Step 2

Solve for $\sqrt{x}$ .

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Step 2.1

Since $\sqrt{x}$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 (5 \sqrt{x} - 9) - 7 (\sqrt{x} - 3) = 3 \sqrt{x}$

Step 2.2

Simplify $4 (5 \sqrt{x} - 9) - 7 (\sqrt{x} - 3)$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$4 (5 \sqrt{x}) + 4 \cdot - 9 - 7 (\sqrt{x} - 3) = 3 \sqrt{x}$

Step 2.2.1.2

Multiply $5$ by $4$ .

$20 \sqrt{x} + 4 \cdot - 9 - 7 (\sqrt{x} - 3) = 3 \sqrt{x}$

Step 2.2.1.3

Multiply $4$ by $- 9$ .

$20 \sqrt{x} - 36 - 7 (\sqrt{x} - 3) = 3 \sqrt{x}$

Step 2.2.1.4

Apply the distributive property.

$20 \sqrt{x} - 36 - 7 \sqrt{x} - 7 \cdot - 3 = 3 \sqrt{x}$

Step 2.2.1.5

Multiply $- 7$ by $- 3$ .

$20 \sqrt{x} - 36 - 7 \sqrt{x} + 21 = 3 \sqrt{x}$

Step 2.2.2

Simplify by adding terms.

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Step 2.2.2.1

Subtract $7 \sqrt{x}$ from $20 \sqrt{x}$ .

$13 \sqrt{x} - 36 + 21 = 3 \sqrt{x}$

Step 2.2.2.2

Add $- 36$ and $21$ .

$13 \sqrt{x} - 15 = 3 \sqrt{x}$

Step 2.3

Move all terms containing $\sqrt{x}$ to the left side of the equation.

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Step 2.3.1

Subtract $3 \sqrt{x}$ from both sides of the equation.

$13 \sqrt{x} - 15 - 3 \sqrt{x} = 0$

Step 2.3.2

Subtract $3 \sqrt{x}$ from $13 \sqrt{x}$ .

$10 \sqrt{x} - 15 = 0$

Step 2.4

Add $15$ to both sides of the equation.

$10 \sqrt{x} = 15$

Step 2.5

Divide each term in $10 \sqrt{x} = 15$ by $10$ and simplify.

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Step 2.5.1

Divide each term in $10 \sqrt{x} = 15$ by $10$ .

$\frac{10 \sqrt{x}}{10} = \frac{15}{10}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Cancel the common factor of $10$ .

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Step 2.5.2.1.1

Cancel the common factor.

$\frac{10 \sqrt{x}}{10} = \frac{15}{10}$

Step 2.5.2.1.2

Divide $\sqrt{x}$ by $1$ .

$\sqrt{x} = \frac{15}{10}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Cancel the common factor of $15$ and $10$ .

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Step 2.5.3.1.1

Factor $5$ out of $15$ .

$\sqrt{x} = \frac{5 (3)}{10}$

Step 2.5.3.1.2

Cancel the common factors.

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Step 2.5.3.1.2.1

Factor $5$ out of $10$ .

$\sqrt{x} = \frac{5 \cdot 3}{5 \cdot 2}$

Step 2.5.3.1.2.2

Cancel the common factor.

$\sqrt{x} = \frac{5 \cdot 3}{5 \cdot 2}$

Step 2.5.3.1.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = {(\frac{3}{2})}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = {(\frac{3}{2})}^{2}$

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(\frac{3}{2})}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(\frac{3}{2})}^{2}$

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Apply the product rule to $\frac{3}{2}$ .

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

Step 4.3.1.2

Raise $3$ to the power of $2$ .

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

Step 4.3.1.3

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

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

Step 5