Basic Math Examples

$t = \frac{1}{32} \cdot (- 17 + \sqrt{14177 - 64 (217 - 17 t - 16 t^{2})})$

Step 1

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

$\frac{1}{32} \cdot (- 17 + \sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}) = t$

Step 2

Solve for $\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}$ .

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Step 2.1

Multiply both sides of the equation by $32$ .

$32 (\frac{1}{32} \cdot (- 17 + \sqrt{14177 - 64 (217 - 17 t - 16 t^{2})})) = 32 t$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $32 (\frac{1}{32} \cdot (- 17 + \sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}))$ .

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Step 2.2.1.1

Apply the distributive property.

$32 (\frac{1}{32} \cdot - 17 + \frac{1}{32} \sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}) = 32 t$

Step 2.2.1.2

Combine $\frac{1}{32}$ and $- 17$ .

$32 (\frac{- 17}{32} + \frac{1}{32} \sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}) = 32 t$

Step 2.2.1.3

Combine $\frac{1}{32}$ and $\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}$ .

$32 (\frac{- 17}{32} + \frac{\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}{32}) = 32 t$

Step 2.2.1.4

Move the negative in front of the fraction.

$32 (- \frac{17}{32} + \frac{\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}{32}) = 32 t$

Step 2.2.1.5

Apply the distributive property.

$32 (- \frac{17}{32}) + 32 \frac{\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}{32} = 32 t$

Step 2.2.1.6

Cancel the common factor of $32$ .

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Step 2.2.1.6.1

Move the leading negative in $- \frac{17}{32}$ into the numerator.

$32 (\frac{- 17}{32}) + 32 \frac{\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}{32} = 32 t$

Step 2.2.1.6.2

Cancel the common factor.

$32 (\frac{- 17}{32}) + 32 \frac{\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}{32} = 32 t$

Step 2.2.1.6.3

Rewrite the expression.

$- 17 + 32 \frac{\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}{32} = 32 t$

Step 2.2.1.7

Cancel the common factor of $32$ .

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Step 2.2.1.7.1

Cancel the common factor.

$- 17 + 32 \frac{\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}{32} = 32 t$

Step 2.2.1.7.2

Rewrite the expression.

$- 17 + \sqrt{14177 - 64 (217 - 17 t - 16 t^{2})} = 32 t$

Step 2.3

Add $17$ to both sides of the equation.

$\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})} = 32 t + 17$

Step 3

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

${\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}}^{2} = {(32 t + 17)}^{2}$

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 $\sqrt{14177 - 64 (217 - 17 t - 16 t^{2})}$ as ${(14177 - 64 (217 - 17 t - 16 t^{2}))}^{\frac{1}{2}}$ .

${({(14177 - 64 (217 - 17 t - 16 t^{2}))}^{\frac{1}{2}})}^{2} = {(32 t + 17)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({(14177 - 64 (217 - 17 t - 16 t^{2}))}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in ${({(14177 - 64 (217 - 17 t - 16 t^{2}))}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

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

${(14177 - 64 (217 - 17 t - 16 t^{2}))}^{\frac{1}{2} \cdot 2} = {(32 t + 17)}^{2}$

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.

${(14177 - 64 (217 - 17 t - 16 t^{2}))}^{\frac{1}{2} \cdot 2} = {(32 t + 17)}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

${(14177 - 64 (217 - 17 t - 16 t^{2}))}^{1} = {(32 t + 17)}^{2}$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Apply the distributive property.

${(14177 - 64 \cdot 217 - 64 (- 17 t) - 64 (- 16 t^{2}))}^{1} = {(32 t + 17)}^{2}$

Step 4.2.1.2.2

Simplify.

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Step 4.2.1.2.2.1

Multiply $- 64$ by $217$ .

${(14177 - 13888 - 64 (- 17 t) - 64 (- 16 t^{2}))}^{1} = {(32 t + 17)}^{2}$

Step 4.2.1.2.2.2

Multiply $- 17$ by $- 64$ .

${(14177 - 13888 + 1088 t - 64 (- 16 t^{2}))}^{1} = {(32 t + 17)}^{2}$

Step 4.2.1.2.2.3

Multiply $- 16$ by $- 64$ .

${(14177 - 13888 + 1088 t + 1024 t^{2})}^{1} = {(32 t + 17)}^{2}$

Step 4.2.1.3

Simplify by subtracting numbers.

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Step 4.2.1.3.1

Subtract $13888$ from $14177$ .

${(289 + 1088 t + 1024 t^{2})}^{1} = {(32 t + 17)}^{2}$

Step 4.2.1.3.2

Simplify.

$289 + 1088 t + 1024 t^{2} = {(32 t + 17)}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(32 t + 17)}^{2}$ .

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Step 4.3.1.1

Rewrite ${(32 t + 17)}^{2}$ as $(32 t + 17) (32 t + 17)$ .

$289 + 1088 t + 1024 t^{2} = (32 t + 17) (32 t + 17)$

Step 4.3.1.2

Expand $(32 t + 17) (32 t + 17)$ using the FOIL Method.

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Step 4.3.1.2.1

Apply the distributive property.

$289 + 1088 t + 1024 t^{2} = 32 t (32 t + 17) + 17 (32 t + 17)$

Step 4.3.1.2.2

Apply the distributive property.

$289 + 1088 t + 1024 t^{2} = 32 t (32 t) + 32 t \cdot 17 + 17 (32 t + 17)$

Step 4.3.1.2.3

Apply the distributive property.

$289 + 1088 t + 1024 t^{2} = 32 t (32 t) + 32 t \cdot 17 + 17 (32 t) + 17 \cdot 17$

Step 4.3.1.3

Simplify and combine like terms.

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Step 4.3.1.3.1

Simplify each term.

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Step 4.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$289 + 1088 t + 1024 t^{2} = 32 \cdot 32 t \cdot t + 32 t \cdot 17 + 17 (32 t) + 17 \cdot 17$

Step 4.3.1.3.1.2

Multiply $t$ by $t$ by adding the exponents.

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Step 4.3.1.3.1.2.1

Move $t$ .

$289 + 1088 t + 1024 t^{2} = 32 \cdot 32 (t \cdot t) + 32 t \cdot 17 + 17 (32 t) + 17 \cdot 17$

Step 4.3.1.3.1.2.2

Multiply $t$ by $t$ .

$289 + 1088 t + 1024 t^{2} = 32 \cdot 32 t^{2} + 32 t \cdot 17 + 17 (32 t) + 17 \cdot 17$

Step 4.3.1.3.1.3

Multiply $32$ by $32$ .

$289 + 1088 t + 1024 t^{2} = 1024 t^{2} + 32 t \cdot 17 + 17 (32 t) + 17 \cdot 17$

Step 4.3.1.3.1.4

Multiply $17$ by $32$ .

$289 + 1088 t + 1024 t^{2} = 1024 t^{2} + 544 t + 17 (32 t) + 17 \cdot 17$

Step 4.3.1.3.1.5

Multiply $32$ by $17$ .

$289 + 1088 t + 1024 t^{2} = 1024 t^{2} + 544 t + 544 t + 17 \cdot 17$

Step 4.3.1.3.1.6

Multiply $17$ by $17$ .

$289 + 1088 t + 1024 t^{2} = 1024 t^{2} + 544 t + 544 t + 289$

Step 4.3.1.3.2

Add $544 t$ and $544 t$ .

$289 + 1088 t + 1024 t^{2} = 1024 t^{2} + 1088 t + 289$

Step 5

Solve for $t$ .

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Step 5.1

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

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Step 5.1.1

Subtract $1024 t^{2}$ from both sides of the equation.

$289 + 1088 t + 1024 t^{2} - 1024 t^{2} = 1088 t + 289$

Step 5.1.2

Subtract $1088 t$ from both sides of the equation.

$289 + 1088 t + 1024 t^{2} - 1024 t^{2} - 1088 t = 289$

Step 5.1.3

Combine the opposite terms in $289 + 1088 t + 1024 t^{2} - 1024 t^{2} - 1088 t$ .

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Step 5.1.3.1

Subtract $1024 t^{2}$ from $1024 t^{2}$ .

$289 + 1088 t + 0 - 1088 t = 289$

Step 5.1.3.2

Add $289 + 1088 t$ and $0$ .

$289 + 1088 t - 1088 t = 289$

Step 5.1.3.3

Subtract $1088 t$ from $1088 t$ .

$289 + 0 = 289$

Step 5.1.3.4

Add $289$ and $0$ .

$289 = 289$

Step 5.2

Since $289 = 289$ , the equation will always be true for any value of $t$ .

All real numbers

Step 6

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$