Algebra Examples

$(x + 3) (x^{2} - 3 x + 9) > (x^{2} - 6) (x - 1)$

Step 1

Simplify $(x + 3) (x^{2} - 3 x + 9)$ .

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Step 1.1

Rewrite.

$0 + 0 + (x + 3) (x^{2} - 3 x + 9) > (x^{2} - 6) (x - 1)$

Step 1.2

Simplify by adding zeros.

$(x + 3) (x^{2} - 3 x + 9) > (x^{2} - 6) (x - 1)$

Step 1.3

Expand $(x + 3) (x^{2} - 3 x + 9)$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{2} + x (- 3 x) + x \cdot 9 + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4

Simplify terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.4.1.1.1

Multiply $x$ by $x^{2}$ .

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Step 1.4.1.1.1.1

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

$x^{1} x^{2} + x (- 3 x) + x \cdot 9 + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.1.1.2

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

$x^{1 + 2} + x (- 3 x) + x \cdot 9 + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.1.2

Add $1$ and $2$ .

$x^{3} + x (- 3 x) + x \cdot 9 + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.2

Rewrite using the commutative property of multiplication.

$x^{3} - 3 x \cdot x + x \cdot 9 + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.3

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

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Step 1.4.1.3.1

Move $x$ .

$x^{3} - 3 (x \cdot x) + x \cdot 9 + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.3.2

Multiply $x$ by $x$ .

$x^{3} - 3 x^{2} + x \cdot 9 + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.4

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

$x^{3} - 3 x^{2} + 9 \cdot x + 3 x^{2} + 3 (- 3 x) + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.5

Multiply $- 3$ by $3$ .

$x^{3} - 3 x^{2} + 9 x + 3 x^{2} - 9 x + 3 \cdot 9 > (x^{2} - 6) (x - 1)$

Step 1.4.1.6

Multiply $3$ by $9$ .

$x^{3} - 3 x^{2} + 9 x + 3 x^{2} - 9 x + 27 > (x^{2} - 6) (x - 1)$

Step 1.4.2

Combine the opposite terms in $x^{3} - 3 x^{2} + 9 x + 3 x^{2} - 9 x + 27$ .

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Step 1.4.2.1

Add $- 3 x^{2}$ and $3 x^{2}$ .

$x^{3} + 9 x + 0 - 9 x + 27 > (x^{2} - 6) (x - 1)$

Step 1.4.2.2

Add $x^{3} + 9 x$ and $0$ .

$x^{3} + 9 x - 9 x + 27 > (x^{2} - 6) (x - 1)$

Step 1.4.2.3

Subtract $9 x$ from $9 x$ .

$x^{3} + 0 + 27 > (x^{2} - 6) (x - 1)$

Step 1.4.2.4

Add $x^{3}$ and $0$ .

$x^{3} + 27 > (x^{2} - 6) (x - 1)$

Step 2

Simplify $(x^{2} - 6) (x - 1)$ .

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Step 2.1

Expand $(x^{2} - 6) (x - 1)$ using the FOIL Method.

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Step 2.1.1

Apply the distributive property.

$x^{3} + 27 > x^{2} (x - 1) - 6 (x - 1)$

Step 2.1.2

Apply the distributive property.

$x^{3} + 27 > x^{2} x + x^{2} \cdot - 1 - 6 (x - 1)$

Step 2.1.3

Apply the distributive property.

$x^{3} + 27 > x^{2} x + x^{2} \cdot - 1 - 6 x - 6 \cdot - 1$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.2.1.1

Multiply $x^{2}$ by $x$ .

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Step 2.2.1.1.1

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

$x^{3} + 27 > x^{2} x^{1} + x^{2} \cdot - 1 - 6 x - 6 \cdot - 1$

Step 2.2.1.1.2

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

$x^{3} + 27 > x^{2 + 1} + x^{2} \cdot - 1 - 6 x - 6 \cdot - 1$

Step 2.2.1.2

Add $2$ and $1$ .

$x^{3} + 27 > x^{3} + x^{2} \cdot - 1 - 6 x - 6 \cdot - 1$

Step 2.2.2

Move $- 1$ to the left of $x^{2}$ .

$x^{3} + 27 > x^{3} - 1 \cdot x^{2} - 6 x - 6 \cdot - 1$

Step 2.2.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$x^{3} + 27 > x^{3} - x^{2} - 6 x - 6 \cdot - 1$

Step 2.2.4

Multiply $- 6$ by $- 1$ .

$x^{3} + 27 > x^{3} - x^{2} - 6 x + 6$

Step 3

Rewrite so $x$ is on the left side of the inequality.

$x^{3} - x^{2} - 6 x + 6 < x^{3} + 27$

Step 4

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

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Step 4.1

Subtract $x^{3}$ from both sides of the inequality.

$x^{3} - x^{2} - 6 x + 6 - x^{3} < 27$

Step 4.2

Combine the opposite terms in $x^{3} - x^{2} - 6 x + 6 - x^{3}$ .

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Step 4.2.1

Subtract $x^{3}$ from $x^{3}$ .

$- x^{2} - 6 x + 6 + 0 < 27$

Step 4.2.2

Add $- x^{2} - 6 x + 6$ and $0$ .

$- x^{2} - 6 x + 6 < 27$

Step 5

Move all terms to the left side of the equation and simplify.

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Step 5.1

Subtract $27$ from both sides of the inequality.

$- x^{2} - 6 x + 6 - 27 < 0$

Step 5.2

Subtract $27$ from $6$ .

$- x^{2} - 6 x - 21 < 0$

Step 6

Convert the inequality to an equation.

$- x^{2} - 6 x - 21 = 0$

Step 7

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8

Substitute the values $a = - 1$ , $b = - 6$ , and $c = - 21$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (- 1 \cdot - 21)}}{2 \cdot - 1}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 21}}{2 \cdot - 1}$

Step 9.1.2

Multiply $- 4 \cdot - 1 \cdot - 21$ .

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Step 9.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{6 \pm \sqrt{36 + 4 \cdot - 21}}{2 \cdot - 1}$

Step 9.1.2.2

Multiply $4$ by $- 21$ .

$x = \frac{6 \pm \sqrt{36 - 84}}{2 \cdot - 1}$

Step 9.1.3

Subtract $84$ from $36$ .

$x = \frac{6 \pm \sqrt{- 48}}{2 \cdot - 1}$

Step 9.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 48}}{2 \cdot - 1}$

Step 9.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot - 1}$

Step 9.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{6 \pm i \cdot \sqrt{48}}{2 \cdot - 1}$

Step 9.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 9.1.7.1

Factor $16$ out of $48$ .

$x = \frac{6 \pm i \cdot \sqrt{16 (3)}}{2 \cdot - 1}$

Step 9.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot - 1}$

Step 9.1.8

Pull terms out from under the radical.

$x = \frac{6 \pm i \cdot (4 \sqrt{3})}{2 \cdot - 1}$

Step 9.1.9

Move $4$ to the left of $i$ .

$x = \frac{6 \pm 4 i \sqrt{3}}{2 \cdot - 1}$

Step 9.2

Multiply $2$ by $- 1$ .

$x = \frac{6 \pm 4 i \sqrt{3}}{- 2}$

Step 9.3

Simplify $\frac{6 \pm 4 i \sqrt{3}}{- 2}$ .

$x = \frac{3 \pm 2 i \sqrt{3}}{- 1}$

Step 9.4

Move the negative one from the denominator of $\frac{3 \pm 2 i \sqrt{3}}{- 1}$ .

$x = - 1 \cdot (3 \pm 2 i \sqrt{3})$

Step 9.5

Rewrite $- 1 \cdot (3 \pm 2 i \sqrt{3})$ as $- (3 \pm 2 i \sqrt{3})$ .

$x = - (3 \pm 2 i \sqrt{3})$

Step 10

Identify the leading coefficient.

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Step 10.1

The leading term in a polynomial is the term with the highest degree.

$- x^{2}$

Step 10.2

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 1$

Step 11

Since there are no real x-intercepts and the leading coefficient is negative, the parabola opens down and $- x^{2} - 6 x + 6$ is always less than $0$ .

All real numbers

Step 12

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$