Basic Math Examples

$\sqrt{10 y^{2} - 4 y - 5} = 3 y$

Step 1

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

${\sqrt{10 y^{2} - 4 y - 5}}^{2} = {(3 y)}^{2}$

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 $\sqrt{10 y^{2} - 4 y - 5}$ as ${(10 y^{2} - 4 y - 5)}^{\frac{1}{2}}$ .

${({(10 y^{2} - 4 y - 5)}^{\frac{1}{2}})}^{2} = {(3 y)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(10 y^{2} - 4 y - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(10 y^{2} - 4 y - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(10 y^{2} - 4 y - 5)}^{\frac{1}{2} \cdot 2} = {(3 y)}^{2}$

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.

${(10 y^{2} - 4 y - 5)}^{\frac{1}{2} \cdot 2} = {(3 y)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(10 y^{2} - 4 y - 5)}^{1} = {(3 y)}^{2}$

Step 2.2.1.2

Simplify.

$10 y^{2} - 4 y - 5 = {(3 y)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Apply the product rule to $3 y$ .

$10 y^{2} - 4 y - 5 = 3^{2} y^{2}$

Step 2.3.1.2

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

$10 y^{2} - 4 y - 5 = 9 y^{2}$

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

$10 y^{2} - 4 y - 5 - 9 y^{2} = 0$

Step 3.1.2

Subtract $9 y^{2}$ from $10 y^{2}$ .

$y^{2} - 4 y - 5 = 0$

Step 3.2

Factor $y^{2} - 4 y - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 3.2.2

Write the factored form using these integers.

$(y - 5) (y + 1) = 0$

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 $0$ .

$y - 5 = 0$

$y + 1 = 0$

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 $y + 1$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $y + 1$ equal to $0$ .

$y + 1 = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$y = - 1$

Step 3.6

The final solution is all the values that make $(y - 5) (y + 1) = 0$ true.

$y = 5, - 1$

Step 4

Exclude the solutions that do not make $\sqrt{10 y^{2} - 4 y - 5} = 3 y$ true.

$y = 5$