Algebra Examples

$2 x^{2} - 3 x + 4 > 0$

Step 1

Convert the inequality to an equation.

$2 x^{2} - 3 x + 4 = 0$

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 $a = 2$ , $b = - 3$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot 4}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 4$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot 4}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 32}}{2 \cdot 2}$

Step 4.1.3

Subtract $32$ from $9$ .

$x = \frac{3 \pm \sqrt{- 23}}{2 \cdot 2}$

Step 4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Step 4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

Step 4.1.6

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

$x = \frac{3 \pm i \sqrt{23}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm i \sqrt{23}}{4}$

Step 5

Identify the leading coefficient.

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Step 5.1

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

$2 x^{2}$

Step 5.2

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

$2$

Step 6

Since there are no real x-intercepts and the leading coefficient is positive, the parabola opens up and $2 x^{2} - 3 x + 4$ is always greater than $0$ .

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$