Algebra Examples

$\sqrt{2 z^{2} - 7 z + 49} = \sqrt{z^{2} + 7 z}$

Step 1

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

${\sqrt{2 z^{2} - 7 z + 49}}^{2} = {\sqrt{z^{2} + 7 z}}^{2}$

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 $\sqrt{2 z^{2} - 7 z + 49}$ as ${(2 z^{2} - 7 z + 49)}^{\frac{1}{2}}$ .

${({(2 z^{2} - 7 z + 49)}^{\frac{1}{2}})}^{2} = {\sqrt{z^{2} + 7 z}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 z^{2} - 7 z + 49)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 z^{2} - 7 z + 49)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(2 z^{2} - 7 z + 49)}^{\frac{1}{2} \cdot 2} = {\sqrt{z^{2} + 7 z}}^{2}$

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.

${(2 z^{2} - 7 z + 49)}^{\frac{1}{2} \cdot 2} = {\sqrt{z^{2} + 7 z}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 z^{2} - 7 z + 49)}^{1} = {\sqrt{z^{2} + 7 z}}^{2}$

Step 2.2.1.2

Simplify.

$2 z^{2} - 7 z + 49 = {\sqrt{z^{2} + 7 z}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{z^{2} + 7 z}}^{2}$ .

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Step 2.3.1.1

Factor $z$ out of $z^{2} + 7 z$ .

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Step 2.3.1.1.1

Factor $z$ out of $z^{2}$ .

$2 z^{2} - 7 z + 49 = {\sqrt{z \cdot z + 7 z}}^{2}$

Step 2.3.1.1.2

Factor $z$ out of $7 z$ .

$2 z^{2} - 7 z + 49 = {\sqrt{z \cdot z + z \cdot 7}}^{2}$

Step 2.3.1.1.3

Factor $z$ out of $z \cdot z + z \cdot 7$ .

$2 z^{2} - 7 z + 49 = {\sqrt{z (z + 7)}}^{2}$

Step 2.3.1.2

Rewrite ${\sqrt{z (z + 7)}}^{2}$ as $z (z + 7)$ .

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Step 2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z (z + 7)}$ as ${(z (z + 7))}^{\frac{1}{2}}$ .

$2 z^{2} - 7 z + 49 = {({(z (z + 7))}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$2 z^{2} - 7 z + 49 = {(z (z + 7))}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$2 z^{2} - 7 z + 49 = {(z (z + 7))}^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$2 z^{2} - 7 z + 49 = {(z (z + 7))}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$2 z^{2} - 7 z + 49 = {(z (z + 7))}^{1}$

Step 2.3.1.2.5

Simplify.

$2 z^{2} - 7 z + 49 = z (z + 7)$

Step 2.3.1.3

Apply the distributive property.

$2 z^{2} - 7 z + 49 = z \cdot z + z \cdot 7$

Step 2.3.1.4

Simplify the expression.

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Step 2.3.1.4.1

Multiply $z$ by $z$ .

$2 z^{2} - 7 z + 49 = z^{2} + z \cdot 7$

Step 2.3.1.4.2

Move $7$ to the left of $z$ .

$2 z^{2} - 7 z + 49 = z^{2} + 7 z$

Step 3

Solve for $z$ .

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Step 3.1

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

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Step 3.1.1

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

$2 z^{2} - 7 z + 49 - z^{2} = 7 z$

Step 3.1.2

Subtract $7 z$ from both sides of the equation.

$2 z^{2} - 7 z + 49 - z^{2} - 7 z = 0$

Step 3.1.3

Subtract $z^{2}$ from $2 z^{2}$ .

$z^{2} - 7 z + 49 - 7 z = 0$

Step 3.1.4

Subtract $7 z$ from $- 7 z$ .

$z^{2} - 14 z + 49 = 0$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

Rewrite $49$ as $7^{2}$ .

$z^{2} - 14 z + 7^{2} = 0$

Step 3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 z = 2 \cdot z \cdot 7$

Step 3.2.3

Rewrite the polynomial.

$z^{2} - 2 \cdot z \cdot 7 + 7^{2} = 0$

Step 3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 7$ .

${(z - 7)}^{2} = 0$

Step 3.3

Set the $z - 7$ equal to $0$ .

$z - 7 = 0$

Step 3.4

Add $7$ to both sides of the equation.

$z = 7$