Basic Math Examples

$a + a + \sqrt{6 + a} - (32 - a) - 98 - \frac{a}{2} = - 19$

Step 1

Solve for $\sqrt{6 + a}$ .

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Step 1.1

Simplify $a + a + \sqrt{6 + a} - (32 - a) - 98 - \frac{a}{2}$ .

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

Apply the distributive property.

$a + a + \sqrt{6 + a} - 1 \cdot 32 - - a - 98 - \frac{a}{2} = - 19$

Step 1.1.1.2

Multiply $- 1$ by $32$ .

$a + a + \sqrt{6 + a} - 32 - - a - 98 - \frac{a}{2} = - 19$

Step 1.1.1.3

Multiply $- - a$ .

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Step 1.1.1.3.1

Multiply $- 1$ by $- 1$ .

$a + a + \sqrt{6 + a} - 32 + 1 a - 98 - \frac{a}{2} = - 19$

Step 1.1.1.3.2

Multiply $a$ by $1$ .

$a + a + \sqrt{6 + a} - 32 + a - 98 - \frac{a}{2} = - 19$

Step 1.1.2

Simplify by adding terms.

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Step 1.1.2.1

Add $a$ and $a$ .

$2 a + \sqrt{6 + a} - 32 + a - 98 - \frac{a}{2} = - 19$

Step 1.1.2.2

Add $2 a$ and $a$ .

$3 a + \sqrt{6 + a} - 32 - 98 - \frac{a}{2} = - 19$

Step 1.1.3

To write $3 a$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{6 + a} - 32 - 98 + 3 a \cdot \frac{2}{2} - \frac{a}{2} = - 19$

Step 1.1.4

Simplify terms.

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Step 1.1.4.1

Combine $3 a$ and $\frac{2}{2}$ .

$\sqrt{6 + a} - 32 - 98 + \frac{3 a \cdot 2}{2} - \frac{a}{2} = - 19$

Step 1.1.4.2

Combine the numerators over the common denominator.

$\sqrt{6 + a} - 32 - 98 + \frac{3 a \cdot 2 - a}{2} = - 19$

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Simplify the numerator.

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Step 1.1.5.1.1

Factor $a$ out of $3 a \cdot 2 - a$ .

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Step 1.1.5.1.1.1

Factor $a$ out of $3 a \cdot 2$ .

$\sqrt{6 + a} - 32 - 98 + \frac{a (3 \cdot 2) - a}{2} = - 19$

Step 1.1.5.1.1.2

Factor $a$ out of $- a$ .

$\sqrt{6 + a} - 32 - 98 + \frac{a (3 \cdot 2) + a \cdot - 1}{2} = - 19$

Step 1.1.5.1.1.3

Factor $a$ out of $a (3 \cdot 2) + a \cdot - 1$ .

$\sqrt{6 + a} - 32 - 98 + \frac{a (3 \cdot 2 - 1)}{2} = - 19$

Step 1.1.5.1.2

Multiply $3$ by $2$ .

$\sqrt{6 + a} - 32 - 98 + \frac{a (6 - 1)}{2} = - 19$

Step 1.1.5.1.3

Subtract $1$ from $6$ .

$\sqrt{6 + a} - 32 - 98 + \frac{a \cdot 5}{2} = - 19$

Step 1.1.5.2

Move $5$ to the left of $a$ .

$\sqrt{6 + a} - 32 - 98 + \frac{5 a}{2} = - 19$

Step 1.1.6

Subtract $98$ from $- 32$ .

$\sqrt{6 + a} - 130 + \frac{5 a}{2} = - 19$

Step 1.2

Move all terms not containing $\sqrt{6 + a}$ to the right side of the equation.

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Step 1.2.1

Add $130$ to both sides of the equation.

$\sqrt{6 + a} + \frac{5 a}{2} = - 19 + 130$

Step 1.2.2

Subtract $\frac{5 a}{2}$ from both sides of the equation.

$\sqrt{6 + a} = - 19 + 130 - \frac{5 a}{2}$

Step 1.2.3

Add $- 19$ and $130$ .

$\sqrt{6 + a} = 111 - \frac{5 a}{2}$

Step 2

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

${\sqrt{6 + a}}^{2} = {(111 - \frac{5 a}{2})}^{2}$

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 $\sqrt{6 + a}$ as ${(6 + a)}^{\frac{1}{2}}$ .

${({(6 + a)}^{\frac{1}{2}})}^{2} = {(111 - \frac{5 a}{2})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(6 + a)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(6 + a)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(6 + a)}^{\frac{1}{2} \cdot 2} = {(111 - \frac{5 a}{2})}^{2}$

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.

${(6 + a)}^{\frac{1}{2} \cdot 2} = {(111 - \frac{5 a}{2})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(6 + a)}^{1} = {(111 - \frac{5 a}{2})}^{2}$

Step 3.2.1.2

Simplify.

$6 + a = {(111 - \frac{5 a}{2})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(111 - \frac{5 a}{2})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(111 - \frac{5 a}{2})}^{2}$ as $(111 - \frac{5 a}{2}) (111 - \frac{5 a}{2})$ .

$6 + a = (111 - \frac{5 a}{2}) (111 - \frac{5 a}{2})$

Step 3.3.1.2

Expand $(111 - \frac{5 a}{2}) (111 - \frac{5 a}{2})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$6 + a = 111 (111 - \frac{5 a}{2}) - \frac{5 a}{2} (111 - \frac{5 a}{2})$

Step 3.3.1.2.2

Apply the distributive property.

$6 + a = 111 \cdot 111 + 111 (- \frac{5 a}{2}) - \frac{5 a}{2} (111 - \frac{5 a}{2})$

Step 3.3.1.2.3

Apply the distributive property.

$6 + a = 111 \cdot 111 + 111 (- \frac{5 a}{2}) - \frac{5 a}{2} \cdot 111 - \frac{5 a}{2} (- \frac{5 a}{2})$

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 $111$ by $111$ .

$6 + a = 12321 + 111 (- \frac{5 a}{2}) - \frac{5 a}{2} \cdot 111 - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.2

Multiply $111 (- \frac{5 a}{2})$ .

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Step 3.3.1.3.1.2.1

Multiply $- 1$ by $111$ .

$6 + a = 12321 - 111 \frac{5 a}{2} - \frac{5 a}{2} \cdot 111 - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.2.2

Combine $- 111$ and $\frac{5 a}{2}$ .

$6 + a = 12321 + \frac{- 111 (5 a)}{2} - \frac{5 a}{2} \cdot 111 - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.2.3

Multiply $5$ by $- 111$ .

$6 + a = 12321 + \frac{- 555 a}{2} - \frac{5 a}{2} \cdot 111 - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.3

Move the negative in front of the fraction.

$6 + a = 12321 - \frac{555 a}{2} - \frac{5 a}{2} \cdot 111 - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.4

Multiply $- \frac{5 a}{2} \cdot 111$ .

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Step 3.3.1.3.1.4.1

Multiply $111$ by $- 1$ .

$6 + a = 12321 - \frac{555 a}{2} - 111 \frac{5 a}{2} - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.4.2

Combine $- 111$ and $\frac{5 a}{2}$ .

$6 + a = 12321 - \frac{555 a}{2} + \frac{- 111 (5 a)}{2} - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.4.3

Multiply $5$ by $- 111$ .

$6 + a = 12321 - \frac{555 a}{2} + \frac{- 555 a}{2} - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.5

Move the negative in front of the fraction.

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} - \frac{5 a}{2} (- \frac{5 a}{2})$

Step 3.3.1.3.1.6

Multiply $- \frac{5 a}{2} (- \frac{5 a}{2})$ .

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Step 3.3.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + 1 \frac{5 a}{2} \frac{5 a}{2}$

Step 3.3.1.3.1.6.2

Multiply $\frac{5 a}{2}$ by $1$ .

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{5 a}{2} \cdot \frac{5 a}{2}$

Step 3.3.1.3.1.6.3

Multiply $\frac{5 a}{2}$ by $\frac{5 a}{2}$ .

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{5 a (5 a)}{2 \cdot 2}$

Step 3.3.1.3.1.6.4

Multiply $5$ by $5$ .

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{25 a \cdot a}{2 \cdot 2}$

Step 3.3.1.3.1.6.5

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

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{25 (a^{1} a)}{2 \cdot 2}$

Step 3.3.1.3.1.6.6

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

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{25 (a^{1} a^{1})}{2 \cdot 2}$

Step 3.3.1.3.1.6.7

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

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{25 a^{1 + 1}}{2 \cdot 2}$

Step 3.3.1.3.1.6.8

Add $1$ and $1$ .

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{25 a^{2}}{2 \cdot 2}$

Step 3.3.1.3.1.6.9

Multiply $2$ by $2$ .

$6 + a = 12321 - \frac{555 a}{2} - \frac{555 a}{2} + \frac{25 a^{2}}{4}$

Step 3.3.1.3.2

Subtract $\frac{555 a}{2}$ from $- \frac{555 a}{2}$ .

$6 + a = 12321 - 2 \frac{555 a}{2} + \frac{25 a^{2}}{4}$

Step 3.3.1.4

Simplify each term.

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Step 3.3.1.4.1

Cancel the common factor of $2$ .

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Step 3.3.1.4.1.1

Factor $2$ out of $- 2$ .

$6 + a = 12321 + 2 (- 1) \frac{555 a}{2} + \frac{25 a^{2}}{4}$

Step 3.3.1.4.1.2

Cancel the common factor.

$6 + a = 12321 + 2 \cdot - 1 \frac{555 a}{2} + \frac{25 a^{2}}{4}$

Step 3.3.1.4.1.3

Rewrite the expression.

$6 + a = 12321 - 1 (555 a) + \frac{25 a^{2}}{4}$

Step 3.3.1.4.2

Multiply $555$ by $- 1$ .

$6 + a = 12321 - 555 a + \frac{25 a^{2}}{4}$

Step 4

Solve for $a$ .

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Step 4.1

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

$12321 - 555 a + \frac{25 a^{2}}{4} = 6 + a$

Step 4.2

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

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Step 4.2.1

Subtract $a$ from both sides of the equation.

$12321 - 555 a + \frac{25 a^{2}}{4} - a = 6$

Step 4.2.2

Subtract $a$ from $- 555 a$ .

$12321 + \frac{25 a^{2}}{4} - 556 a = 6$

Step 4.3

Move all terms to the left side of the equation and simplify.

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Step 4.3.1

Subtract $6$ from both sides of the equation.

$12321 + \frac{25 a^{2}}{4} - 556 a - 6 = 0$

Step 4.3.2

Subtract $6$ from $12321$ .

$\frac{25 a^{2}}{4} - 556 a + 12315 = 0$

Step 4.4

Multiply through by the least common denominator $4$ , then simplify.

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Step 4.4.1

Apply the distributive property.

$4 (\frac{25 a^{2}}{4}) + 4 (- 556 a) + 4 \cdot 12315 = 0$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Cancel the common factor of $4$ .

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Step 4.4.2.1.1

Cancel the common factor.

$4 (\frac{25 a^{2}}{4}) + 4 (- 556 a) + 4 \cdot 12315 = 0$

Step 4.4.2.1.2

Rewrite the expression.

$25 a^{2} + 4 (- 556 a) + 4 \cdot 12315 = 0$

Step 4.4.2.2

Multiply $- 556$ by $4$ .

$25 a^{2} - 2224 a + 4 \cdot 12315 = 0$

Step 4.4.2.3

Multiply $4$ by $12315$ .

$25 a^{2} - 2224 a + 49260 = 0$

Step 4.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.6

Substitute the values $a = 25$ , $b = - 2224$ , and $c = 49260$ into the quadratic formula and solve for $a$ .

$\frac{2224 \pm \sqrt{{(- 2224)}^{2} - 4 \cdot (25 \cdot 49260)}}{2 \cdot 25}$

Step 4.7

Simplify.

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Raise $- 2224$ to the power of $2$ .

$a = \frac{2224 \pm \sqrt{4946176 - 4 \cdot 25 \cdot 49260}}{2 \cdot 25}$

Step 4.7.1.2

Multiply $- 4 \cdot 25 \cdot 49260$ .

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Step 4.7.1.2.1

Multiply $- 4$ by $25$ .

$a = \frac{2224 \pm \sqrt{4946176 - 100 \cdot 49260}}{2 \cdot 25}$

Step 4.7.1.2.2

Multiply $- 100$ by $49260$ .

$a = \frac{2224 \pm \sqrt{4946176 - 4926000}}{2 \cdot 25}$

Step 4.7.1.3

Subtract $4926000$ from $4946176$ .

$a = \frac{2224 \pm \sqrt{20176}}{2 \cdot 25}$

Step 4.7.1.4

Rewrite $20176$ as $4^{2} \cdot 1261$ .

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Step 4.7.1.4.1

Factor $16$ out of $20176$ .

$a = \frac{2224 \pm \sqrt{16 (1261)}}{2 \cdot 25}$

Step 4.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$a = \frac{2224 \pm \sqrt{4^{2} \cdot 1261}}{2 \cdot 25}$

Step 4.7.1.5

Pull terms out from under the radical.

$a = \frac{2224 \pm 4 \sqrt{1261}}{2 \cdot 25}$

Step 4.7.2

Multiply $2$ by $25$ .

$a = \frac{2224 \pm 4 \sqrt{1261}}{50}$

Step 4.7.3

Simplify $\frac{2224 \pm 4 \sqrt{1261}}{50}$ .

$a = \frac{1112 \pm 2 \sqrt{1261}}{25}$

Step 4.8

The final answer is the combination of both solutions.

$a = \frac{1112 + 2 \sqrt{1261}}{25}, \frac{1112 - 2 \sqrt{1261}}{25}$

Step 5

Exclude the solutions that do not make $a + a + \sqrt{6 + a} - (32 - a) - 98 - \frac{a}{2} = - 19$ true.

$a = \frac{1112 - 2 \sqrt{1261}}{25}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$a = \frac{1112 - 2 \sqrt{1261}}{25}$

Decimal Form:

$a = 41.63915505 \dots$