Algebra Examples

$30 x - 45 = 5 x^{2}$

Step 1

Subtract $5 x^{2}$ from both sides of the equation.

$30 x - 45 - 5 x^{2} = 0$

Step 2

Factor $5$ out of $30 x - 45 - 5 x^{2}$ .

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Step 2.1

Factor $5$ out of $30 x$ .

$5 (6 x) - 45 - 5 x^{2} = 0$

Step 2.2

Factor $5$ out of $- 45$ .

$5 (6 x) + 5 (- 9) - 5 x^{2} = 0$

Step 2.3

Factor $5$ out of $- 5 x^{2}$ .

$5 (6 x) + 5 (- 9) + 5 (- x^{2}) = 0$

Step 2.4

Factor $5$ out of $5 (6 x) + 5 (- 9)$ .

$5 (6 x - 9) + 5 (- x^{2}) = 0$

Step 2.5

Factor $5$ out of $5 (6 x - 9) + 5 (- x^{2})$ .

$5 (6 x - 9 - x^{2}) = 0$

Step 3

Factor.

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Step 3.1

Factor by grouping.

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Step 3.1.1

Reorder terms.

$5 (- x^{2} + 6 x - 9) = 0$

Step 3.1.2

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 $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 6$ .

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Step 3.1.2.1

Factor $6$ out of $6 x$ .

$5 (- x^{2} + 6 (x) - 9) = 0$

Step 3.1.2.2

Rewrite $6$ as $3$ plus $3$

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

Step 3.1.2.3

Apply the distributive property.

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

Step 3.1.3

Factor out the greatest common factor from each group.

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Step 3.1.3.1

Group the first two terms and the last two terms.

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

Step 3.1.3.2

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

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

Step 3.1.4

Factor the polynomial by factoring out the greatest common factor, $- x + 3$ .

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

Step 3.2

Remove unnecessary parentheses.

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

Step 4

Combine exponents.

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Step 4.1

Factor $- 1$ out of $- x$ .

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

Step 4.2

Rewrite $3$ as $- 1 (- 3)$ .

$5 (- (x) - 1 \cdot - 3) (x - 3) = 0$

Step 4.3

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 4.4

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$5 (- 1 (x - 3)) (x - 3) = 0$

Step 4.5

Remove parentheses.

$5 \cdot (- 1 (x - 3) (x - 3)) = 0$

Step 4.6

Raise $x - 3$ to the power of $1$ .

$5 \cdot (- 1 ((x - 3) (x - 3))) = 0$

Step 4.7

Raise $x - 3$ to the power of $1$ .

$5 \cdot (- 1 ((x - 3) (x - 3))) = 0$

Step 4.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 \cdot (- 1 {(x - 3)}^{1 + 1}) = 0$

Step 4.9

Add $1$ and $1$ .

$5 \cdot (- 1 {(x - 3)}^{2}) = 0$

Step 4.10

Multiply $5$ by $- 1$ .

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

Step 5

Divide each term in $- 5 {(x - 3)}^{2} = 0$ by $- 5$ and simplify.

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Step 5.1

Divide each term in $- 5 {(x - 3)}^{2} = 0$ by $- 5$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 5$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide ${(x - 3)}^{2}$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 5$ .

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

Step 6

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

$x - 3 = 0$

Step 7

Add $3$ to both sides of the equation.

$x = 3$