Basic Math Examples

$\frac{3 + 4 s}{2} = 2 \sqrt{1 + s}$

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.

$2 \sqrt{1 + s} = \frac{3 + 4 s}{2}$

Step 2

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

${(2 \sqrt{1 + s})}^{2} = {(\frac{3 + 4 s}{2})}^{2}$

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

${(2 {(1 + s)}^{\frac{1}{2}})}^{2} = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $2 {(1 + s)}^{\frac{1}{2}}$ .

$2^{2} {({(1 + s)}^{\frac{1}{2}})}^{2} = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.2.1.2

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

$4 {({(1 + s)}^{\frac{1}{2}})}^{2} = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.2.1.3

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

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Step 3.2.1.3.1

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

$4 {(1 + s)}^{\frac{1}{2} \cdot 2} = {(\frac{3 + 4 s}{2})}^{2}$

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.

$4 {(1 + s)}^{\frac{1}{2} \cdot 2} = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.2.1.3.2.2

Rewrite the expression.

$4 {(1 + s)}^{1} = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.2.1.4

Simplify.

$4 (1 + s) = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.2.1.5

Apply the distributive property.

$4 \cdot 1 + 4 s = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.2.1.6

Multiply $4$ by $1$ .

$4 + 4 s = {(\frac{3 + 4 s}{2})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\frac{3 + 4 s}{2})}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $\frac{3 + 4 s}{2}$ .

$4 + 4 s = \frac{{(3 + 4 s)}^{2}}{2^{2}}$

Step 3.3.1.2

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

$4 + 4 s = \frac{{(3 + 4 s)}^{2}}{4}$

Step 4

Solve for $s$ .

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Step 4.1

Multiply both sides by $4$ .

$(4 + 4 s) \cdot 4 = \frac{{(3 + 4 s)}^{2}}{4} \cdot 4$

Step 4.2

Simplify.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Simplify $(4 + 4 s) \cdot 4$ .

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Step 4.2.1.1.1

Apply the distributive property.

$4 \cdot 4 + 4 s \cdot 4 = \frac{{(3 + 4 s)}^{2}}{4} \cdot 4$

Step 4.2.1.1.2

Simplify the expression.

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Step 4.2.1.1.2.1

Multiply $4$ by $4$ .

$16 + 4 s \cdot 4 = \frac{{(3 + 4 s)}^{2}}{4} \cdot 4$

Step 4.2.1.1.2.2

Multiply $4$ by $4$ .

$16 + 16 s = \frac{{(3 + 4 s)}^{2}}{4} \cdot 4$

Step 4.2.1.1.2.3

Reorder $16$ and $16 s$ .

$16 s + 16 = \frac{{(3 + 4 s)}^{2}}{4} \cdot 4$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

$16 s + 16 = \frac{{(3 + 4 s)}^{2}}{4} \cdot 4$

Step 4.2.2.1.2

Rewrite the expression.

$16 s + 16 = {(3 + 4 s)}^{2}$

Step 4.3

Solve for $s$ .

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Step 4.3.1

Simplify ${(3 + 4 s)}^{2}$ .

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Step 4.3.1.1

Rewrite.

$16 s + 16 = 0 + 0 + {(3 + 4 s)}^{2}$

Step 4.3.1.2

Rewrite ${(3 + 4 s)}^{2}$ as $(3 + 4 s) (3 + 4 s)$ .

$16 s + 16 = (3 + 4 s) (3 + 4 s)$

Step 4.3.1.3

Expand $(3 + 4 s) (3 + 4 s)$ using the FOIL Method.

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Step 4.3.1.3.1

Apply the distributive property.

$16 s + 16 = 3 (3 + 4 s) + 4 s (3 + 4 s)$

Step 4.3.1.3.2

Apply the distributive property.

$16 s + 16 = 3 \cdot 3 + 3 (4 s) + 4 s (3 + 4 s)$

Step 4.3.1.3.3

Apply the distributive property.

$16 s + 16 = 3 \cdot 3 + 3 (4 s) + 4 s \cdot 3 + 4 s (4 s)$

Step 4.3.1.4

Simplify and combine like terms.

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Step 4.3.1.4.1

Simplify each term.

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Step 4.3.1.4.1.1

Multiply $3$ by $3$ .

$16 s + 16 = 9 + 3 (4 s) + 4 s \cdot 3 + 4 s (4 s)$

Step 4.3.1.4.1.2

Multiply $4$ by $3$ .

$16 s + 16 = 9 + 12 s + 4 s \cdot 3 + 4 s (4 s)$

Step 4.3.1.4.1.3

Multiply $3$ by $4$ .

$16 s + 16 = 9 + 12 s + 12 s + 4 s (4 s)$

Step 4.3.1.4.1.4

Rewrite using the commutative property of multiplication.

$16 s + 16 = 9 + 12 s + 12 s + 4 \cdot 4 s \cdot s$

Step 4.3.1.4.1.5

Multiply $s$ by $s$ by adding the exponents.

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Step 4.3.1.4.1.5.1

Move $s$ .

$16 s + 16 = 9 + 12 s + 12 s + 4 \cdot 4 (s \cdot s)$

Step 4.3.1.4.1.5.2

Multiply $s$ by $s$ .

$16 s + 16 = 9 + 12 s + 12 s + 4 \cdot 4 s^{2}$

Step 4.3.1.4.1.6

Multiply $4$ by $4$ .

$16 s + 16 = 9 + 12 s + 12 s + 16 s^{2}$

Step 4.3.1.4.2

Add $12 s$ and $12 s$ .

$16 s + 16 = 9 + 24 s + 16 s^{2}$

Step 4.3.2

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

$9 + 24 s + 16 s^{2} = 16 s + 16$

Step 4.3.3

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

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Step 4.3.3.1

Subtract $16 s$ from both sides of the equation.

$9 + 24 s + 16 s^{2} - 16 s = 16$

Step 4.3.3.2

Subtract $16 s$ from $24 s$ .

$9 + 16 s^{2} + 8 s = 16$

Step 4.3.4

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

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Step 4.3.4.1

Subtract $16$ from both sides of the equation.

$9 + 16 s^{2} + 8 s - 16 = 0$

Step 4.3.4.2

Subtract $16$ from $9$ .

$16 s^{2} + 8 s - 7 = 0$

Step 4.3.5

Use the quadratic formula to find the solutions.

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

Step 4.3.6

Substitute the values $a = 16$ , $b = 8$ , and $c = - 7$ into the quadratic formula and solve for $s$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (16 \cdot - 7)}}{2 \cdot 16}$

Step 4.3.7

Simplify.

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Step 4.3.7.1

Simplify the numerator.

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Step 4.3.7.1.1

Raise $8$ to the power of $2$ .

$s = \frac{- 8 \pm \sqrt{64 - 4 \cdot 16 \cdot - 7}}{2 \cdot 16}$

Step 4.3.7.1.2

Multiply $- 4 \cdot 16 \cdot - 7$ .

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Step 4.3.7.1.2.1

Multiply $- 4$ by $16$ .

$s = \frac{- 8 \pm \sqrt{64 - 64 \cdot - 7}}{2 \cdot 16}$

Step 4.3.7.1.2.2

Multiply $- 64$ by $- 7$ .

$s = \frac{- 8 \pm \sqrt{64 + 448}}{2 \cdot 16}$

Step 4.3.7.1.3

Add $64$ and $448$ .

$s = \frac{- 8 \pm \sqrt{512}}{2 \cdot 16}$

Step 4.3.7.1.4

Rewrite $512$ as $16^{2} \cdot 2$ .

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Step 4.3.7.1.4.1

Factor $256$ out of $512$ .

$s = \frac{- 8 \pm \sqrt{256 (2)}}{2 \cdot 16}$

Step 4.3.7.1.4.2

Rewrite $256$ as $16^{2}$ .

$s = \frac{- 8 \pm \sqrt{16^{2} \cdot 2}}{2 \cdot 16}$

Step 4.3.7.1.5

Pull terms out from under the radical.

$s = \frac{- 8 \pm 16 \sqrt{2}}{2 \cdot 16}$

Step 4.3.7.2

Multiply $2$ by $16$ .

$s = \frac{- 8 \pm 16 \sqrt{2}}{32}$

Step 4.3.7.3

Simplify $\frac{- 8 \pm 16 \sqrt{2}}{32}$ .

$s = \frac{- 1 \pm 2 \sqrt{2}}{4}$

Step 4.3.8

The final answer is the combination of both solutions.

$s = - \frac{1 - 2 \sqrt{2}}{4}, - \frac{1 + 2 \sqrt{2}}{4}$

Step 5

Exclude the solutions that do not make $\frac{3 + 4 s}{2} = 2 \sqrt{1 + s}$ true.

$s = - \frac{1 - 2 \sqrt{2}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$s = - \frac{1 - 2 \sqrt{2}}{4}$

Decimal Form:

$s = 0.45710678 \dots$