Finite Math Examples

$\sqrt{2 x + 1} - \sqrt{9 + 4 x} = - 2$

Step 1

Add $\sqrt{9 + 4 x}$ to both sides of the equation.

$\sqrt{2 x + 1} = - 2 + \sqrt{9 + 4 x}$

Step 2

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

${\sqrt{2 x + 1}}^{2} = {(- 2 + \sqrt{9 + 4 x})}^{2}$

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 $\sqrt{2 x + 1}$ as ${(2 x + 1)}^{\frac{1}{2}}$ .

${({(2 x + 1)}^{\frac{1}{2}})}^{2} = {(- 2 + \sqrt{9 + 4 x})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(2 x + 1)}^{\frac{1}{2} \cdot 2} = {(- 2 + \sqrt{9 + 4 x})}^{2}$

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.

${(2 x + 1)}^{\frac{1}{2} \cdot 2} = {(- 2 + \sqrt{9 + 4 x})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x + 1)}^{1} = {(- 2 + \sqrt{9 + 4 x})}^{2}$

Step 3.2.1.2

Simplify.

$2 x + 1 = {(- 2 + \sqrt{9 + 4 x})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(- 2 + \sqrt{9 + 4 x})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(- 2 + \sqrt{9 + 4 x})}^{2}$ as $(- 2 + \sqrt{9 + 4 x}) (- 2 + \sqrt{9 + 4 x})$ .

$2 x + 1 = (- 2 + \sqrt{9 + 4 x}) (- 2 + \sqrt{9 + 4 x})$

Step 3.3.1.2

Expand $(- 2 + \sqrt{9 + 4 x}) (- 2 + \sqrt{9 + 4 x})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$2 x + 1 = - 2 (- 2 + \sqrt{9 + 4 x}) + \sqrt{9 + 4 x} (- 2 + \sqrt{9 + 4 x})$

Step 3.3.1.2.2

Apply the distributive property.

$2 x + 1 = - 2 \cdot - 2 - 2 \sqrt{9 + 4 x} + \sqrt{9 + 4 x} (- 2 + \sqrt{9 + 4 x})$

Step 3.3.1.2.3

Apply the distributive property.

$2 x + 1 = - 2 \cdot - 2 - 2 \sqrt{9 + 4 x} + \sqrt{9 + 4 x} \cdot - 2 + \sqrt{9 + 4 x} \sqrt{9 + 4 x}$

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

Multiply $- 2$ by $- 2$ .

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} + \sqrt{9 + 4 x} \cdot - 2 + \sqrt{9 + 4 x} \sqrt{9 + 4 x}$

Step 3.3.1.3.1.2

Move $- 2$ to the left of $\sqrt{9 + 4 x}$ .

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \cdot \sqrt{9 + 4 x} + \sqrt{9 + 4 x} \sqrt{9 + 4 x}$

Step 3.3.1.3.1.3

Multiply $\sqrt{9 + 4 x} \sqrt{9 + 4 x}$ .

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Step 3.3.1.3.1.3.1

Raise $\sqrt{9 + 4 x}$ to the power of $1$ .

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {\sqrt{9 + 4 x}}^{1} \sqrt{9 + 4 x}$

Step 3.3.1.3.1.3.2

Raise $\sqrt{9 + 4 x}$ to the power of $1$ .

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {\sqrt{9 + 4 x}}^{1} {\sqrt{9 + 4 x}}^{1}$

Step 3.3.1.3.1.3.3

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

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {\sqrt{9 + 4 x}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {\sqrt{9 + 4 x}}^{2}$

Step 3.3.1.3.1.4

Rewrite ${\sqrt{9 + 4 x}}^{2}$ as $9 + 4 x$ .

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Step 3.3.1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + 4 x}$ as ${(9 + 4 x)}^{\frac{1}{2}}$ .

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {({(9 + 4 x)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {(9 + 4 x)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {(9 + 4 x)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {(9 + 4 x)}^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + {(9 + 4 x)}^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$2 x + 1 = 4 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + 9 + 4 x$

Step 3.3.1.3.2

Add $4$ and $9$ .

$2 x + 1 = 13 - 2 \sqrt{9 + 4 x} - 2 \sqrt{9 + 4 x} + 4 x$

Step 3.3.1.3.3

Subtract $2 \sqrt{9 + 4 x}$ from $- 2 \sqrt{9 + 4 x}$ .

$2 x + 1 = 13 - 4 \sqrt{9 + 4 x} + 4 x$

Step 4

Solve for $- 4 \sqrt{9 + 4 x}$ .

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Step 4.1

Rewrite the equation as $13 - 4 \sqrt{9 + 4 x} + 4 x = 2 x + 1$ .

$13 - 4 \sqrt{9 + 4 x} + 4 x = 2 x + 1$

Step 4.2

Move all terms not containing $- 4 \sqrt{9 + 4 x}$ to the right side of the equation.

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Step 4.2.1

Subtract $13$ from both sides of the equation.

$- 4 \sqrt{9 + 4 x} + 4 x = 2 x + 1 - 13$

Step 4.2.2

Subtract $4 x$ from both sides of the equation.

$- 4 \sqrt{9 + 4 x} = 2 x + 1 - 13 - 4 x$

Step 4.2.3

Subtract $4 x$ from $2 x$ .

$- 4 \sqrt{9 + 4 x} = - 2 x + 1 - 13$

Step 4.2.4

Subtract $13$ from $1$ .

$- 4 \sqrt{9 + 4 x} = - 2 x - 12$

Step 5

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

${(- 4 \sqrt{9 + 4 x})}^{2} = {(- 2 x - 12)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 + 4 x}$ as ${(9 + 4 x)}^{\frac{1}{2}}$ .

${(- 4 {(9 + 4 x)}^{\frac{1}{2}})}^{2} = {(- 2 x - 12)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 4 {(9 + 4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 4 {(9 + 4 x)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(9 + 4 x)}^{\frac{1}{2}})}^{2} = {(- 2 x - 12)}^{2}$

Step 6.2.1.2

Raise $- 4$ to the power of $2$ .

$16 {({(9 + 4 x)}^{\frac{1}{2}})}^{2} = {(- 2 x - 12)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(9 + 4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$16 {(9 + 4 x)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 12)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$16 {(9 + 4 x)}^{\frac{1}{2} \cdot 2} = {(- 2 x - 12)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 {(9 + 4 x)}^{1} = {(- 2 x - 12)}^{2}$

Step 6.2.1.4

Simplify.

$16 (9 + 4 x) = {(- 2 x - 12)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$16 \cdot 9 + 16 (4 x) = {(- 2 x - 12)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $16$ by $9$ .

$144 + 16 (4 x) = {(- 2 x - 12)}^{2}$

Step 6.2.1.6.2

Multiply $4$ by $16$ .

$144 + 64 x = {(- 2 x - 12)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(- 2 x - 12)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(- 2 x - 12)}^{2}$ as $(- 2 x - 12) (- 2 x - 12)$ .

$144 + 64 x = (- 2 x - 12) (- 2 x - 12)$

Step 6.3.1.2

Expand $(- 2 x - 12) (- 2 x - 12)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$144 + 64 x = - 2 x (- 2 x - 12) - 12 (- 2 x - 12)$

Step 6.3.1.2.2

Apply the distributive property.

$144 + 64 x = - 2 x (- 2 x) - 2 x \cdot - 12 - 12 (- 2 x - 12)$

Step 6.3.1.2.3

Apply the distributive property.

$144 + 64 x = - 2 x (- 2 x) - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$144 + 64 x = - 2 \cdot - 2 x \cdot x - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 6.3.1.3.1.2

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

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Step 6.3.1.3.1.2.1

Move $x$ .

$144 + 64 x = - 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$144 + 64 x = - 2 \cdot - 2 x^{2} - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 6.3.1.3.1.3

Multiply $- 2$ by $- 2$ .

$144 + 64 x = 4 x^{2} - 2 x \cdot - 12 - 12 (- 2 x) - 12 \cdot - 12$

Step 6.3.1.3.1.4

Multiply $- 12$ by $- 2$ .

$144 + 64 x = 4 x^{2} + 24 x - 12 (- 2 x) - 12 \cdot - 12$

Step 6.3.1.3.1.5

Multiply $- 2$ by $- 12$ .

$144 + 64 x = 4 x^{2} + 24 x + 24 x - 12 \cdot - 12$

Step 6.3.1.3.1.6

Multiply $- 12$ by $- 12$ .

$144 + 64 x = 4 x^{2} + 24 x + 24 x + 144$

Step 6.3.1.3.2

Add $24 x$ and $24 x$ .

$144 + 64 x = 4 x^{2} + 48 x + 144$

Step 7

Solve for $x$ .

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Step 7.1

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

$4 x^{2} + 48 x + 144 = 144 + 64 x$

Step 7.2

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

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Step 7.2.1

Subtract $64 x$ from both sides of the equation.

$4 x^{2} + 48 x + 144 - 64 x = 144$

Step 7.2.2

Subtract $64 x$ from $48 x$ .

$4 x^{2} - 16 x + 144 = 144$

Step 7.3

Subtract $144$ from both sides of the equation.

$4 x^{2} - 16 x + 144 - 144 = 0$

Step 7.4

Combine the opposite terms in $4 x^{2} - 16 x + 144 - 144$ .

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Step 7.4.1

Subtract $144$ from $144$ .

$4 x^{2} - 16 x + 0 = 0$

Step 7.4.2

Add $4 x^{2} - 16 x$ and $0$ .

$4 x^{2} - 16 x = 0$

Step 7.5

Factor $4 x$ out of $4 x^{2} - 16 x$ .

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Step 7.5.1

Factor $4 x$ out of $4 x^{2}$ .

$4 x (x) - 16 x = 0$

Step 7.5.2

Factor $4 x$ out of $- 16 x$ .

$4 x (x) + 4 x (- 4) = 0$

Step 7.5.3

Factor $4 x$ out of $4 x (x) + 4 x (- 4)$ .

$4 x (x - 4) = 0$

Step 7.6

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

$x = 0$

$x - 4 = 0$

Step 7.7

Set $x$ equal to $0$ .

$x = 0$

Step 7.8

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 7.8.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 7.8.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7.9

The final solution is all the values that make $4 x (x - 4) = 0$ true.

$x = 0, 4$