Basic Math Examples

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

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

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}

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

Step 2

Simplify each side of the equation.

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Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{10 y^{2} - 4 y - 5}

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

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

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

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

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

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Step 2.2.1.1

Multiply the exponents in

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

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

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Step 2.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.2.1.1.2

Cancel the common factor of

2

$2$ .

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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.

$y = 5$