Finite Math Examples

$\sqrt{5 y^{2} - 8 y + 15} = 2 y$

Step 1

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

${\sqrt{5 y^{2} - 8 y + 15}}^{2} = {(2 y)}^{2}$

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 $\sqrt{5 y^{2} - 8 y + 15}$ as ${(5 y^{2} - 8 y + 15)}^{\frac{1}{2}}$ .

${({(5 y^{2} - 8 y + 15)}^{\frac{1}{2}})}^{2} = {(2 y)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(5 y^{2} - 8 y + 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(5 y^{2} - 8 y + 15)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(5 y^{2} - 8 y + 15)}^{\frac{1}{2} \cdot 2} = {(2 y)}^{2}$

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.

${(5 y^{2} - 8 y + 15)}^{\frac{1}{2} \cdot 2} = {(2 y)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(5 y^{2} - 8 y + 15)}^{1} = {(2 y)}^{2}$

Step 2.2.1.2

Simplify.

$5 y^{2} - 8 y + 15 = {(2 y)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the product rule to $2 y$ .

$5 y^{2} - 8 y + 15 = 2^{2} y^{2}$

Step 2.3.1.2

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

$5 y^{2} - 8 y + 15 = 4 y^{2}$

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4 y^{2}$ from both sides of the equation.

$5 y^{2} - 8 y + 15 - 4 y^{2} = 0$

Step 3.1.2

Subtract $4 y^{2}$ from $5 y^{2}$ .

$y^{2} - 8 y + 15 = 0$

Step 3.2

Factor $y^{2} - 8 y + 15$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 3.2.2

Write the factored form using these integers.

$(y - 5) (y - 3) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 5 = 0$

$y - 3 = 0$

Step 3.4

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 3.4.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 3.4.2

Add $5$ to both sides of the equation.

$y = 5$

Step 3.5

Set $y - 3$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $y - 3$ equal to $0$ .

$y - 3 = 0$

Step 3.5.2

Add $3$ to both sides of the equation.

$y = 3$

Step 3.6

The final solution is all the values that make $(y - 5) (y - 3) = 0$ true.

$y = 5, 3$