Trigonometry Examples

$25 x^{2} - 30 x + 9 = {(5 x - 3)}^{2}$

Step 1

Simplify ${(5 x - 3)}^{2}$ .

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Step 1.1

Rewrite.

$25 x^{2} - 30 x + 9 = 0 + 0 + {(5 x - 3)}^{2}$

Step 1.2

Rewrite ${(5 x - 3)}^{2}$ as $(5 x - 3) (5 x - 3)$ .

$25 x^{2} - 30 x + 9 = (5 x - 3) (5 x - 3)$

Step 1.3

Expand $(5 x - 3) (5 x - 3)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$25 x^{2} - 30 x + 9 = 5 x (5 x - 3) - 3 (5 x - 3)$

Step 1.3.2

Apply the distributive property.

$25 x^{2} - 30 x + 9 = 5 x (5 x) + 5 x \cdot - 3 - 3 (5 x - 3)$

Step 1.3.3

Apply the distributive property.

$25 x^{2} - 30 x + 9 = 5 x (5 x) + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$25 x^{2} - 30 x + 9 = 5 \cdot 5 x \cdot x + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3$

Step 1.4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.2.1

Move $x$ .

$25 x^{2} - 30 x + 9 = 5 \cdot 5 (x \cdot x) + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3$

Step 1.4.1.2.2

Multiply $x$ by $x$ .

$25 x^{2} - 30 x + 9 = 5 \cdot 5 x^{2} + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3$

Step 1.4.1.3

Multiply $5$ by $5$ .

$25 x^{2} - 30 x + 9 = 25 x^{2} + 5 x \cdot - 3 - 3 (5 x) - 3 \cdot - 3$

Step 1.4.1.4

Multiply $- 3$ by $5$ .

$25 x^{2} - 30 x + 9 = 25 x^{2} - 15 x - 3 (5 x) - 3 \cdot - 3$

Step 1.4.1.5

Multiply $5$ by $- 3$ .

$25 x^{2} - 30 x + 9 = 25 x^{2} - 15 x - 15 x - 3 \cdot - 3$

Step 1.4.1.6

Multiply $- 3$ by $- 3$ .

$25 x^{2} - 30 x + 9 = 25 x^{2} - 15 x - 15 x + 9$

Step 1.4.2

Subtract $15 x$ from $- 15 x$ .

$25 x^{2} - 30 x + 9 = 25 x^{2} - 30 x + 9$

Step 2

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

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Step 2.1

Subtract $25 x^{2}$ from both sides of the equation.

$25 x^{2} - 30 x + 9 - 25 x^{2} = - 30 x + 9$

Step 2.2

Add $30 x$ to both sides of the equation.

$25 x^{2} - 30 x + 9 - 25 x^{2} + 30 x = 9$

Step 2.3

Combine the opposite terms in $25 x^{2} - 30 x + 9 - 25 x^{2} + 30 x$ .

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Step 2.3.1

Subtract $25 x^{2}$ from $25 x^{2}$ .

$- 30 x + 9 + 0 + 30 x = 9$

Step 2.3.2

Add $- 30 x + 9$ and $0$ .

$- 30 x + 9 + 30 x = 9$

Step 2.3.3

Add $- 30 x$ and $30 x$ .

$0 + 9 = 9$

Step 2.3.4

Add $0$ and $9$ .

$9 = 9$

Step 3

Since $9 = 9$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$