Linear Algebra Examples

$3 x^{2} - 2 x + 1 < 0$

Step 1

Convert the inequality to an equation.

$3 x^{2} - 2 x + 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 3$ , $b = - 2$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot 1)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot 1}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 3}$

Step 4.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 3}$

Step 4.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 3}$

Step 4.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 3}$

Step 4.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 3}$

Step 4.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.1.7.1

Factor $4$ out of $8$ .

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

Step 4.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 3}$

Step 4.1.8

Pull terms out from under the radical.

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

Step 4.1.9

Move $2$ to the left of $i$ .

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

Step 4.2

Multiply $2$ by $3$ .

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

Step 4.3

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

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

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot 1}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 3}$

Step 5.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 3}$

Step 5.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 3}$

Step 5.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 3}$

Step 5.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 3}$

Step 5.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.1.7.1

Factor $4$ out of $8$ .

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

Step 5.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 3}$

Step 5.1.8

Pull terms out from under the radical.

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

Step 5.1.9

Move $2$ to the left of $i$ .

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

Step 5.2

Multiply $2$ by $3$ .

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

Step 5.3

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

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

Step 5.4

Change the $\pm$ to $+$ .

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

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot 1}}{2 \cdot 3}$

Step 6.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 12}}{2 \cdot 3}$

Step 6.1.3

Subtract $12$ from $4$ .

$x = \frac{2 \pm \sqrt{- 8}}{2 \cdot 3}$

Step 6.1.4

Rewrite $- 8$ as $- 1 (8)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 8}}{2 \cdot 3}$

Step 6.1.5

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{8}}{2 \cdot 3}$

Step 6.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{8}}{2 \cdot 3}$

Step 6.1.7

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.1.7.1

Factor $4$ out of $8$ .

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

Step 6.1.7.2

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

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 2}}{2 \cdot 3}$

Step 6.1.8

Pull terms out from under the radical.

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

Step 6.1.9

Move $2$ to the left of $i$ .

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

Step 6.2

Multiply $2$ by $3$ .

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

Step 6.3

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

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

Step 6.4

Change the $\pm$ to $-$ .

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

Step 7

Identify the leading coefficient.

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Step 7.1

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

$3 x^{2}$

Step 7.2

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

$3$

Step 8

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $3 x^{2} - 2 x + 1$ is always greater than $0$ .

No solution