Basic Math Examples

$3 \sqrt{2 d - 3} + 2 \sqrt{7 - d} = 11$

Step 1

Subtract $2 \sqrt{7 - d}$ from both sides of the equation.

$3 \sqrt{2 d - 3} = 11 - 2 \sqrt{7 - d}$

Step 2

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

${(3 \sqrt{2 d - 3})}^{2} = {(11 - 2 \sqrt{7 - d})}^{2}$

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 $\sqrt{2 d - 3}$ as ${(2 d - 3)}^{\frac{1}{2}}$ .

${(3 {(2 d - 3)}^{\frac{1}{2}})}^{2} = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(3 {(2 d - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $3 {(2 d - 3)}^{\frac{1}{2}}$ .

$3^{2} {({(2 d - 3)}^{\frac{1}{2}})}^{2} = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.2

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

$9 {({(2 d - 3)}^{\frac{1}{2}})}^{2} = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.3

Multiply the exponents in ${({(2 d - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.3.1

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

$9 {(2 d - 3)}^{\frac{1}{2} \cdot 2} = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.2.1.3.2.1

Cancel the common factor.

$9 {(2 d - 3)}^{\frac{1}{2} \cdot 2} = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$9 {(2 d - 3)}^{1} = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.4

Simplify.

$9 (2 d - 3) = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.5

Apply the distributive property.

$9 (2 d) + 9 \cdot - 3 = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.6

Multiply.

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Step 3.2.1.6.1

Multiply $2$ by $9$ .

$18 d + 9 \cdot - 3 = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.2.1.6.2

Multiply $9$ by $- 3$ .

$18 d - 27 = {(11 - 2 \sqrt{7 - d})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(11 - 2 \sqrt{7 - d})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(11 - 2 \sqrt{7 - d})}^{2}$ as $(11 - 2 \sqrt{7 - d}) (11 - 2 \sqrt{7 - d})$ .

$18 d - 27 = (11 - 2 \sqrt{7 - d}) (11 - 2 \sqrt{7 - d})$

Step 3.3.1.2

Expand $(11 - 2 \sqrt{7 - d}) (11 - 2 \sqrt{7 - d})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$18 d - 27 = 11 (11 - 2 \sqrt{7 - d}) - 2 \sqrt{7 - d} (11 - 2 \sqrt{7 - d})$

Step 3.3.1.2.2

Apply the distributive property.

$18 d - 27 = 11 \cdot 11 + 11 (- 2 \sqrt{7 - d}) - 2 \sqrt{7 - d} (11 - 2 \sqrt{7 - d})$

Step 3.3.1.2.3

Apply the distributive property.

$18 d - 27 = 11 \cdot 11 + 11 (- 2 \sqrt{7 - d}) - 2 \sqrt{7 - d} \cdot 11 - 2 \sqrt{7 - d} (- 2 \sqrt{7 - d})$

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 $11$ by $11$ .

$18 d - 27 = 121 + 11 (- 2 \sqrt{7 - d}) - 2 \sqrt{7 - d} \cdot 11 - 2 \sqrt{7 - d} (- 2 \sqrt{7 - d})$

Step 3.3.1.3.1.2

Multiply $- 2$ by $11$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 2 \sqrt{7 - d} \cdot 11 - 2 \sqrt{7 - d} (- 2 \sqrt{7 - d})$

Step 3.3.1.3.1.3

Multiply $11$ by $- 2$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} - 2 \sqrt{7 - d} (- 2 \sqrt{7 - d})$

Step 3.3.1.3.1.4

Multiply $- 2 \sqrt{7 - d} (- 2 \sqrt{7 - d})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 2$ by $- 2$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 \sqrt{7 - d} \sqrt{7 - d}$

Step 3.3.1.3.1.4.2

Raise $\sqrt{7 - d}$ to the power of $1$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 ({\sqrt{7 - d}}^{1} \sqrt{7 - d})$

Step 3.3.1.3.1.4.3

Raise $\sqrt{7 - d}$ to the power of $1$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 ({\sqrt{7 - d}}^{1} {\sqrt{7 - d}}^{1})$

Step 3.3.1.3.1.4.4

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

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 {\sqrt{7 - d}}^{1 + 1}$

Step 3.3.1.3.1.4.5

Add $1$ and $1$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 {\sqrt{7 - d}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{7 - d}}^{2}$ as $7 - d$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 - d}$ as ${(7 - d)}^{\frac{1}{2}}$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 {({(7 - d)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 {(7 - d)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 {(7 - d)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 {(7 - d)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 {(7 - d)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 (7 - d)$

Step 3.3.1.3.1.6

Apply the distributive property.

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 4 \cdot 7 + 4 (- d)$

Step 3.3.1.3.1.7

Multiply $4$ by $7$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 28 + 4 (- d)$

Step 3.3.1.3.1.8

Multiply $- 1$ by $4$ .

$18 d - 27 = 121 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} + 28 - 4 d$

Step 3.3.1.3.2

Add $121$ and $28$ .

$18 d - 27 = 149 - 22 \sqrt{7 - d} - 22 \sqrt{7 - d} - 4 d$

Step 3.3.1.3.3

Subtract $22 \sqrt{7 - d}$ from $- 22 \sqrt{7 - d}$ .

$18 d - 27 = 149 - 44 \sqrt{7 - d} - 4 d$

Step 4

Solve for $- 44 \sqrt{7 - d}$ .

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Step 4.1

Rewrite the equation as $149 - 44 \sqrt{7 - d} - 4 d = 18 d - 27$ .

$149 - 44 \sqrt{7 - d} - 4 d = 18 d - 27$

Step 4.2

Move all terms not containing $- 44 \sqrt{7 - d}$ to the right side of the equation.

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Step 4.2.1

Subtract $149$ from both sides of the equation.

$- 44 \sqrt{7 - d} - 4 d = 18 d - 27 - 149$

Step 4.2.2

Add $4 d$ to both sides of the equation.

$- 44 \sqrt{7 - d} = 18 d - 27 - 149 + 4 d$

Step 4.2.3

Add $18 d$ and $4 d$ .

$- 44 \sqrt{7 - d} = 22 d - 27 - 149$

Step 4.2.4

Subtract $149$ from $- 27$ .

$- 44 \sqrt{7 - d} = 22 d - 176$

Step 5

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

${(- 44 \sqrt{7 - d})}^{2} = {(22 d - 176)}^{2}$

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 $\sqrt{7 - d}$ as ${(7 - d)}^{\frac{1}{2}}$ .

${(- 44 {(7 - d)}^{\frac{1}{2}})}^{2} = {(22 d - 176)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 44 {(7 - d)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 44 {(7 - d)}^{\frac{1}{2}}$ .

${(- 44)}^{2} {({(7 - d)}^{\frac{1}{2}})}^{2} = {(22 d - 176)}^{2}$

Step 6.2.1.2

Raise $- 44$ to the power of $2$ .

$1936 {({(7 - d)}^{\frac{1}{2}})}^{2} = {(22 d - 176)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(7 - d)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$1936 {(7 - d)}^{\frac{1}{2} \cdot 2} = {(22 d - 176)}^{2}$

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.

$1936 {(7 - d)}^{\frac{1}{2} \cdot 2} = {(22 d - 176)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$1936 {(7 - d)}^{1} = {(22 d - 176)}^{2}$

Step 6.2.1.4

Simplify.

$1936 (7 - d) = {(22 d - 176)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$1936 \cdot 7 + 1936 (- d) = {(22 d - 176)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $1936$ by $7$ .

$13552 + 1936 (- d) = {(22 d - 176)}^{2}$

Step 6.2.1.6.2

Multiply $- 1$ by $1936$ .

$13552 - 1936 d = {(22 d - 176)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(22 d - 176)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(22 d - 176)}^{2}$ as $(22 d - 176) (22 d - 176)$ .

$13552 - 1936 d = (22 d - 176) (22 d - 176)$

Step 6.3.1.2

Expand $(22 d - 176) (22 d - 176)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$13552 - 1936 d = 22 d (22 d - 176) - 176 (22 d - 176)$

Step 6.3.1.2.2

Apply the distributive property.

$13552 - 1936 d = 22 d (22 d) + 22 d \cdot - 176 - 176 (22 d - 176)$

Step 6.3.1.2.3

Apply the distributive property.

$13552 - 1936 d = 22 d (22 d) + 22 d \cdot - 176 - 176 (22 d) - 176 \cdot - 176$

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.

$13552 - 1936 d = 22 \cdot 22 d \cdot d + 22 d \cdot - 176 - 176 (22 d) - 176 \cdot - 176$

Step 6.3.1.3.1.2

Multiply $d$ by $d$ by adding the exponents.

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Step 6.3.1.3.1.2.1

Move $d$ .

$13552 - 1936 d = 22 \cdot 22 (d \cdot d) + 22 d \cdot - 176 - 176 (22 d) - 176 \cdot - 176$

Step 6.3.1.3.1.2.2

Multiply $d$ by $d$ .

$13552 - 1936 d = 22 \cdot 22 d^{2} + 22 d \cdot - 176 - 176 (22 d) - 176 \cdot - 176$

Step 6.3.1.3.1.3

Multiply $22$ by $22$ .

$13552 - 1936 d = 484 d^{2} + 22 d \cdot - 176 - 176 (22 d) - 176 \cdot - 176$

Step 6.3.1.3.1.4

Multiply $- 176$ by $22$ .

$13552 - 1936 d = 484 d^{2} - 3872 d - 176 (22 d) - 176 \cdot - 176$

Step 6.3.1.3.1.5

Multiply $22$ by $- 176$ .

$13552 - 1936 d = 484 d^{2} - 3872 d - 3872 d - 176 \cdot - 176$

Step 6.3.1.3.1.6

Multiply $- 176$ by $- 176$ .

$13552 - 1936 d = 484 d^{2} - 3872 d - 3872 d + 30976$

Step 6.3.1.3.2

Subtract $3872 d$ from $- 3872 d$ .

$13552 - 1936 d = 484 d^{2} - 7744 d + 30976$

Step 7

Solve for $d$ .

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Step 7.1

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

$484 d^{2} - 7744 d + 30976 = 13552 - 1936 d$

Step 7.2

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

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Step 7.2.1

Add $1936 d$ to both sides of the equation.

$484 d^{2} - 7744 d + 30976 + 1936 d = 13552$

Step 7.2.2

Add $- 7744 d$ and $1936 d$ .

$484 d^{2} - 5808 d + 30976 = 13552$

Step 7.3

Subtract $13552$ from both sides of the equation.

$484 d^{2} - 5808 d + 30976 - 13552 = 0$

Step 7.4

Subtract $13552$ from $30976$ .

$484 d^{2} - 5808 d + 17424 = 0$

Step 7.5

Factor the left side of the equation.

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Step 7.5.1

Factor $484$ out of $484 d^{2} - 5808 d + 17424$ .

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Step 7.5.1.1

Factor $484$ out of $484 d^{2}$ .

$484 (d^{2}) - 5808 d + 17424 = 0$

Step 7.5.1.2

Factor $484$ out of $- 5808 d$ .

$484 (d^{2}) + 484 (- 12 d) + 17424 = 0$

Step 7.5.1.3

Factor $484$ out of $17424$ .

$484 d^{2} + 484 (- 12 d) + 484 \cdot 36 = 0$

Step 7.5.1.4

Factor $484$ out of $484 d^{2} + 484 (- 12 d)$ .

$484 (d^{2} - 12 d) + 484 \cdot 36 = 0$

Step 7.5.1.5

Factor $484$ out of $484 (d^{2} - 12 d) + 484 \cdot 36$ .

$484 (d^{2} - 12 d + 36) = 0$

Step 7.5.2

Factor using the perfect square rule.

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Step 7.5.2.1

Rewrite $36$ as $6^{2}$ .

$484 (d^{2} - 12 d + 6^{2}) = 0$

Step 7.5.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 d = 2 \cdot d \cdot 6$

Step 7.5.2.3

Rewrite the polynomial.

$484 (d^{2} - 2 \cdot d \cdot 6 + 6^{2}) = 0$

Step 7.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = d$ and $b = 6$ .

$484 {(d - 6)}^{2} = 0$

Step 7.6

Divide each term in $484 {(d - 6)}^{2} = 0$ by $484$ and simplify.

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Step 7.6.1

Divide each term in $484 {(d - 6)}^{2} = 0$ by $484$ .

$\frac{484 {(d - 6)}^{2}}{484} = \frac{0}{484}$

Step 7.6.2

Simplify the left side.

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Step 7.6.2.1

Cancel the common factor of $484$ .

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Step 7.6.2.1.1

Cancel the common factor.

$\frac{484 {(d - 6)}^{2}}{484} = \frac{0}{484}$

Step 7.6.2.1.2

Divide ${(d - 6)}^{2}$ by $1$ .

${(d - 6)}^{2} = \frac{0}{484}$

Step 7.6.3

Simplify the right side.

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Step 7.6.3.1

Divide $0$ by $484$ .

${(d - 6)}^{2} = 0$

Step 7.7

Set the $d - 6$ equal to $0$ .

$d - 6 = 0$

Step 7.8

Add $6$ to both sides of the equation.

$d = 6$