Finite Math Examples

$3 y = \sqrt{9 y^{2} - 4 y + 24}$

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.

$\sqrt{9 y^{2} - 4 y + 24} = 3 y$

Step 2

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

${\sqrt{9 y^{2} - 4 y + 24}}^{2} = {(3 y)}^{2}$

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 $\sqrt{9 y^{2} - 4 y + 24}$ as ${(9 y^{2} - 4 y + 24)}^{\frac{1}{2}}$ .

${({(9 y^{2} - 4 y + 24)}^{\frac{1}{2}})}^{2} = {(3 y)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(9 y^{2} - 4 y + 24)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(9 y^{2} - 4 y + 24)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(9 y^{2} - 4 y + 24)}^{\frac{1}{2} \cdot 2} = {(3 y)}^{2}$

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.

${(9 y^{2} - 4 y + 24)}^{\frac{1}{2} \cdot 2} = {(3 y)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(9 y^{2} - 4 y + 24)}^{1} = {(3 y)}^{2}$

Step 3.2.1.2

Simplify.

$9 y^{2} - 4 y + 24 = {(3 y)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 y)}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $3 y$ .

$9 y^{2} - 4 y + 24 = 3^{2} y^{2}$

Step 3.3.1.2

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

$9 y^{2} - 4 y + 24 = 9 y^{2}$

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

Subtract $9 y^{2}$ from both sides of the equation.

$9 y^{2} - 4 y + 24 - 9 y^{2} = 0$

Step 4.1.2

Combine the opposite terms in $9 y^{2} - 4 y + 24 - 9 y^{2}$ .

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Step 4.1.2.1

Subtract $9 y^{2}$ from $9 y^{2}$ .

$- 4 y + 24 + 0 = 0$

Step 4.1.2.2

Add $- 4 y + 24$ and $0$ .

$- 4 y + 24 = 0$

Step 4.2

Subtract $24$ from both sides of the equation.

$- 4 y = - 24$

Step 4.3

Divide each term in $- 4 y = - 24$ by $- 4$ and simplify.

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Step 4.3.1

Divide each term in $- 4 y = - 24$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- 24}{- 4}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 4$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 24}{- 4}$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 24}{- 4}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $- 24$ by $- 4$ .

$y = 6$