Finite Math Examples

$\sqrt{6 x^{2} + 16 x} = \sqrt{3 x + 35}$

Step 1

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

${\sqrt{6 x^{2} + 16 x}}^{2} = {\sqrt{3 x + 35}}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x^{2} + 16 x}$ as ${(6 x^{2} + 16 x)}^{\frac{1}{2}}$ .

${({(6 x^{2} + 16 x)}^{\frac{1}{2}})}^{2} = {\sqrt{3 x + 35}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(6 x^{2} + 16 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(6 x^{2} + 16 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(6 x^{2} + 16 x)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 x + 35}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(6 x^{2} + 16 x)}^{\frac{1}{2} \cdot 2} = {\sqrt{3 x + 35}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(6 x^{2} + 16 x)}^{1} = {\sqrt{3 x + 35}}^{2}$

Step 2.2.1.2

Simplify.

$6 x^{2} + 16 x = {\sqrt{3 x + 35}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{3 x + 35}}^{2}$ as $3 x + 35$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x + 35}$ as ${(3 x + 35)}^{\frac{1}{2}}$ .

$6 x^{2} + 16 x = {({(3 x + 35)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$6 x^{2} + 16 x = {(3 x + 35)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$6 x^{2} + 16 x = {(3 x + 35)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$6 x^{2} + 16 x = {(3 x + 35)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$6 x^{2} + 16 x = {(3 x + 35)}^{1}$

Step 2.3.1.5

Simplify.

$6 x^{2} + 16 x = 3 x + 35$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $3 x$ from both sides of the equation.

$6 x^{2} + 16 x - 3 x = 35$

Step 3.1.2

Subtract $3 x$ from $16 x$ .

$6 x^{2} + 13 x = 35$

Step 3.2

Subtract $35$ from both sides of the equation.

$6 x^{2} + 13 x - 35 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = 6$ , $b = 13$ , and $c = - 35$ into the quadratic formula and solve for $x$ .

$\frac{- 13 \pm \sqrt{13^{2} - 4 \cdot (6 \cdot - 35)}}{2 \cdot 6}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $13$ to the power of $2$ .

$x = \frac{- 13 \pm \sqrt{169 - 4 \cdot 6 \cdot - 35}}{2 \cdot 6}$

Step 3.5.1.2

Multiply $- 4 \cdot 6 \cdot - 35$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $6$ .

$x = \frac{- 13 \pm \sqrt{169 - 24 \cdot - 35}}{2 \cdot 6}$

Step 3.5.1.2.2

Multiply $- 24$ by $- 35$ .

$x = \frac{- 13 \pm \sqrt{169 + 840}}{2 \cdot 6}$

Step 3.5.1.3

Add $169$ and $840$ .

$x = \frac{- 13 \pm \sqrt{1009}}{2 \cdot 6}$

Step 3.5.2

Multiply $2$ by $6$ .

$x = \frac{- 13 \pm \sqrt{1009}}{12}$

Step 3.6

The final answer is the combination of both solutions.

$x = - \frac{13 - \sqrt{1009}}{12}, - \frac{13 + \sqrt{1009}}{12}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{13 - \sqrt{1009}}{12}, - \frac{13 + \sqrt{1009}}{12}$

Decimal Form:

$x = 1.56373002 \dots, - 3.73039669 \dots$