Algebra Examples

$x^{2} \geq 4 (x - 5)$

Step 1

Simplify $4 (x - 5)$ .

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Step 1.1

Apply the distributive property.

$x^{2} \geq 4 x + 4 \cdot - 5$

Step 1.2

Multiply $4$ by $- 5$ .

$x^{2} \geq 4 x - 20$

Step 2

Subtract $4 x$ from both sides of the inequality.

$x^{2} - 4 x \geq - 20$

Step 3

Add $20$ to both sides of the inequality.

$x^{2} - 4 x + 20 \geq 0$

Step 4

Convert the inequality to an equation.

$x^{2} - 4 x + 20 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 4$ , and $c = 20$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 20)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 20$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 20}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $20$ .

$x = \frac{4 \pm \sqrt{16 - 80}}{2 \cdot 1}$

Step 7.1.3

Subtract $80$ from $16$ .

$x = \frac{4 \pm \sqrt{- 64}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

Step 7.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 7.1.7

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 8}{2 \cdot 1}$

Step 7.1.9

Move $8$ to the left of $i$ .

$x = \frac{4 \pm 8 i}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 8 i}{2}$

Step 7.3

Simplify $\frac{4 \pm 8 i}{2}$ .

$x = 2 \pm 4 i$

Step 8

Identify the leading coefficient.

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Step 8.1

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

$x^{2}$

Step 8.2

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

$1$

Step 9

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

All real numbers

Step 10

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$