Basic Math Examples

$3 y - 4 = \sqrt{y^{2} - 3 y + 6}$

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.

$\sqrt{y^{2} - 3 y + 6} = 3 y - 4$

Step 2

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

${\sqrt{y^{2} - 3 y + 6}}^{2} = {(3 y - 4)}^{2}$

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 $\sqrt{y^{2} - 3 y + 6}$ as ${(y^{2} - 3 y + 6)}^{\frac{1}{2}}$ .

${({(y^{2} - 3 y + 6)}^{\frac{1}{2}})}^{2} = {(3 y - 4)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(y^{2} - 3 y + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(y^{2} - 3 y + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(y^{2} - 3 y + 6)}^{\frac{1}{2} \cdot 2} = {(3 y - 4)}^{2}$

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.

${(y^{2} - 3 y + 6)}^{\frac{1}{2} \cdot 2} = {(3 y - 4)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(y^{2} - 3 y + 6)}^{1} = {(3 y - 4)}^{2}$

Step 3.2.1.2

Simplify.

$y^{2} - 3 y + 6 = {(3 y - 4)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 y - 4)}^{2}$ .

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Step 3.3.1.1

Rewrite ${(3 y - 4)}^{2}$ as $(3 y - 4) (3 y - 4)$ .

$y^{2} - 3 y + 6 = (3 y - 4) (3 y - 4)$

Step 3.3.1.2

Expand $(3 y - 4) (3 y - 4)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$y^{2} - 3 y + 6 = 3 y (3 y - 4) - 4 (3 y - 4)$

Step 3.3.1.2.2

Apply the distributive property.

$y^{2} - 3 y + 6 = 3 y (3 y) + 3 y \cdot - 4 - 4 (3 y - 4)$

Step 3.3.1.2.3

Apply the distributive property.

$y^{2} - 3 y + 6 = 3 y (3 y) + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$y^{2} - 3 y + 6 = 3 \cdot 3 y \cdot y + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4$

Step 3.3.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.3.1.3.1.2.1

Move $y$ .

$y^{2} - 3 y + 6 = 3 \cdot 3 (y \cdot y) + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4$

Step 3.3.1.3.1.2.2

Multiply $y$ by $y$ .

$y^{2} - 3 y + 6 = 3 \cdot 3 y^{2} + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4$

Step 3.3.1.3.1.3

Multiply $3$ by $3$ .

$y^{2} - 3 y + 6 = 9 y^{2} + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4$

Step 3.3.1.3.1.4

Multiply $- 4$ by $3$ .

$y^{2} - 3 y + 6 = 9 y^{2} - 12 y - 4 (3 y) - 4 \cdot - 4$

Step 3.3.1.3.1.5

Multiply $3$ by $- 4$ .

$y^{2} - 3 y + 6 = 9 y^{2} - 12 y - 12 y - 4 \cdot - 4$

Step 3.3.1.3.1.6

Multiply $- 4$ by $- 4$ .

$y^{2} - 3 y + 6 = 9 y^{2} - 12 y - 12 y + 16$

Step 3.3.1.3.2

Subtract $12 y$ from $- 12 y$ .

$y^{2} - 3 y + 6 = 9 y^{2} - 24 y + 16$

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.

$y^{2} - 3 y + 6 - 9 y^{2} = - 24 y + 16$

Step 4.1.2

Add $24 y$ to both sides of the equation.

$y^{2} - 3 y + 6 - 9 y^{2} + 24 y = 16$

Step 4.1.3

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

$- 8 y^{2} - 3 y + 6 + 24 y = 16$

Step 4.1.4

Add $- 3 y$ and $24 y$ .

$- 8 y^{2} + 21 y + 6 = 16$

Step 4.2

Subtract $16$ from both sides of the equation.

$- 8 y^{2} + 21 y + 6 - 16 = 0$

Step 4.3

Subtract $16$ from $6$ .

$- 8 y^{2} + 21 y - 10 = 0$

Step 4.4

Factor the left side of the equation.

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Step 4.4.1

Factor $- 1$ out of $- 8 y^{2} + 21 y - 10$ .

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Step 4.4.1.1

Factor $- 1$ out of $- 8 y^{2}$ .

$- (8 y^{2}) + 21 y - 10 = 0$

Step 4.4.1.2

Factor $- 1$ out of $21 y$ .

$- (8 y^{2}) - (- 21 y) - 10 = 0$

Step 4.4.1.3

Rewrite $- 10$ as $- 1 (10)$ .

$- (8 y^{2}) - (- 21 y) - 1 \cdot 10 = 0$

Step 4.4.1.4

Factor $- 1$ out of $- (8 y^{2}) - (- 21 y)$ .

$- (8 y^{2} - 21 y) - 1 \cdot 10 = 0$

Step 4.4.1.5

Factor $- 1$ out of $- (8 y^{2} - 21 y) - 1 (10)$ .

$- (8 y^{2} - 21 y + 10) = 0$

Step 4.4.2

Factor.

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Step 4.4.2.1

Factor by grouping.

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Step 4.4.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 8 \cdot 10 = 80$ and whose sum is $b = - 21$ .

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Step 4.4.2.1.1.1

Factor $- 21$ out of $- 21 y$ .

$- (8 y^{2} - 21 y + 10) = 0$

Step 4.4.2.1.1.2

Rewrite $- 21$ as $- 5$ plus $- 16$

$- (8 y^{2} + (- 5 - 16) y + 10) = 0$

Step 4.4.2.1.1.3

Apply the distributive property.

$- (8 y^{2} - 5 y - 16 y + 10) = 0$

Step 4.4.2.1.2

Factor out the greatest common factor from each group.

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Step 4.4.2.1.2.1

Group the first two terms and the last two terms.

$- ((8 y^{2} - 5 y) - 16 y + 10) = 0$

Step 4.4.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (y (8 y - 5) - 2 (8 y - 5)) = 0$

Step 4.4.2.1.3

Factor the polynomial by factoring out the greatest common factor, $8 y - 5$ .

$- ((8 y - 5) (y - 2)) = 0$

Step 4.4.2.2

Remove unnecessary parentheses.

$- (8 y - 5) (y - 2) = 0$

Step 4.5

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

$8 y - 5 = 0$

$y - 2 = 0$

Step 4.6

Set $8 y - 5$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $8 y - 5$ equal to $0$ .

$8 y - 5 = 0$

Step 4.6.2

Solve $8 y - 5 = 0$ for $y$ .

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Step 4.6.2.1

Add $5$ to both sides of the equation.

$8 y = 5$

Step 4.6.2.2

Divide each term in $8 y = 5$ by $8$ and simplify.

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Step 4.6.2.2.1

Divide each term in $8 y = 5$ by $8$ .

$\frac{8 y}{8} = \frac{5}{8}$

Step 4.6.2.2.2

Simplify the left side.

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Step 4.6.2.2.2.1

Cancel the common factor of $8$ .

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Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{5}{8}$

Step 4.6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{5}{8}$

Step 4.7

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 4.7.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 4.7.2

Add $2$ to both sides of the equation.

$y = 2$

Step 4.8

The final solution is all the values that make $- (8 y - 5) (y - 2) = 0$ true.

$y = \frac{5}{8}, 2$

Step 5

Exclude the solutions that do not make $3 y - 4 = \sqrt{y^{2} - 3 y + 6}$ true.

$y = 2$