Algebra Examples

$10 x^{2} - 25 = x^{2}$

Step 1

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

$10 x^{2} - 25 - x^{2} = 0$

Step 2

Subtract $x^{2}$ from $10 x^{2}$ .

$9 x^{2} - 25 = 0$

Step 3

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

${(3 x)}^{2} - 25 = 0$

Step 4

Rewrite $25$ as $5^{2}$ .

${(3 x)}^{2} - 5^{2} = 0$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 5$ .

$(3 x + 5) (3 x - 5) = 0$

Step 6

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

$3 x + 5 = 0$

$3 x - 5 = 0$

Step 7

Set $3 x + 5$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Step 7.2

Solve $3 x + 5 = 0$ for $x$ .

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Step 7.2.1

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Step 7.2.2

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 7.2.2.1

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 8

Set $3 x - 5$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Step 8.2

Solve $3 x - 5 = 0$ for $x$ .

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Step 8.2.1

Add $5$ to both sides of the equation.

$3 x = 5$

Step 8.2.2

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 8.2.2.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 9

The final solution is all the values that make $(3 x + 5) (3 x - 5) = 0$ true.

$x = - \frac{5}{3}, \frac{5}{3}$