Algebra Examples

$\sqrt{25 - 3 x} = x - 9$

Step 1

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

${\sqrt{25 - 3 x}}^{2} = {(x - 9)}^{2}$

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 $\sqrt{25 - 3 x}$ as ${(25 - 3 x)}^{\frac{1}{2}}$ .

${({(25 - 3 x)}^{\frac{1}{2}})}^{2} = {(x - 9)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(25 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(25 - 3 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(25 - 3 x)}^{\frac{1}{2} \cdot 2} = {(x - 9)}^{2}$

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.

${(25 - 3 x)}^{\frac{1}{2} \cdot 2} = {(x - 9)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(25 - 3 x)}^{1} = {(x - 9)}^{2}$

Step 2.2.1.2

Simplify.

$25 - 3 x = {(x - 9)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x - 9)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(x - 9)}^{2}$ as $(x - 9) (x - 9)$ .

$25 - 3 x = (x - 9) (x - 9)$

Step 2.3.1.2

Expand $(x - 9) (x - 9)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$25 - 3 x = x (x - 9) - 9 (x - 9)$

Step 2.3.1.2.2

Apply the distributive property.

$25 - 3 x = x \cdot x + x \cdot - 9 - 9 (x - 9)$

Step 2.3.1.2.3

Apply the distributive property.

$25 - 3 x = x \cdot x + x \cdot - 9 - 9 x - 9 \cdot - 9$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

$25 - 3 x = x^{2} + x \cdot - 9 - 9 x - 9 \cdot - 9$

Step 2.3.1.3.1.2

Move $- 9$ to the left of $x$ .

$25 - 3 x = x^{2} - 9 \cdot x - 9 x - 9 \cdot - 9$

Step 2.3.1.3.1.3

Multiply $- 9$ by $- 9$ .

$25 - 3 x = x^{2} - 9 x - 9 x + 81$

Step 2.3.1.3.2

Subtract $9 x$ from $- 9 x$ .

$25 - 3 x = x^{2} - 18 x + 81$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 18 x + 81 = 25 - 3 x$

Step 3.2

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

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Step 3.2.1

Add $3 x$ to both sides of the equation.

$x^{2} - 18 x + 81 + 3 x = 25$

Step 3.2.2

Add $- 18 x$ and $3 x$ .

$x^{2} - 15 x + 81 = 25$

Step 3.3

Subtract $25$ from both sides of the equation.

$x^{2} - 15 x + 81 - 25 = 0$

Step 3.4

Subtract $25$ from $81$ .

$x^{2} - 15 x + 56 = 0$

Step 3.5

Factor $x^{2} - 15 x + 56$ using the AC method.

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Step 3.5.1

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 $56$ and whose sum is $- 15$ .

$- 8, - 7$

Step 3.5.2

Write the factored form using these integers.

$(x - 8) (x - 7) = 0$

Step 3.6

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

$x - 8 = 0$

$x - 7 = 0$

Step 3.7

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 3.7.2

Add $8$ to both sides of the equation.

$x = 8$

Step 3.8

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 3.8.2

Add $7$ to both sides of the equation.

$x = 7$

Step 3.9

The final solution is all the values that make $(x - 8) (x - 7) = 0$ true.

$x = 8, 7$

Step 4

Exclude the solutions that do not make $\sqrt{25 - 3 x} = x - 9$ true.

$No solution$