Algebra Examples

$x^{2} + c x + 5 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = c$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- c \pm \sqrt{c^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 3.1.1

Multiply $- 4$ by $1$ .

$x = \frac{- c \pm \sqrt{c^{2} - 4 \cdot 5}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $5$ .

$x = \frac{- c \pm \sqrt{c^{2} - 20}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{- c \pm \sqrt{c^{2} - 20}}{2}$

Step 4

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

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Step 4.1

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 4.1.1

Multiply $- 4$ by $1$ .

$x = \frac{- c \pm \sqrt{c^{2} - 4 \cdot 5}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $5$ .

$x = \frac{- c \pm \sqrt{c^{2} - 20}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- c \pm \sqrt{c^{2} - 20}}{2}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{- c + \sqrt{c^{2} - 20}}{2}$

Step 4.4

Factor $- 1$ out of $- c$ .

$x = \frac{- (c) + \sqrt{c^{2} - 20}}{2}$

Step 4.5

Factor $- 1$ out of $\sqrt{c^{2} - 20}$ .

$x = \frac{- (c) - 1 (- \sqrt{c^{2} - 20})}{2}$

Step 4.6

Factor $- 1$ out of $- (c) - 1 (- \sqrt{c^{2} - 20})$ .

$x = \frac{- (c - \sqrt{c^{2} - 20})}{2}$

Step 4.7

Rewrite $- (c - \sqrt{c^{2} - 20})$ as $- 1 (c - \sqrt{c^{2} - 20})$ .

$x = \frac{- 1 (c - \sqrt{c^{2} - 20})}{2}$

Step 4.8

Move the negative in front of the fraction.

$x = - \frac{c - \sqrt{c^{2} - 20}}{2}$

Step 5

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

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Step 5.1

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 5.1.1

Multiply $- 4$ by $1$ .

$x = \frac{- c \pm \sqrt{c^{2} - 4 \cdot 5}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $5$ .

$x = \frac{- c \pm \sqrt{c^{2} - 20}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- c \pm \sqrt{c^{2} - 20}}{2}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{- c - \sqrt{c^{2} - 20}}{2}$

Step 5.4

Factor $- 1$ out of $- c$ .

$x = \frac{- (c) - \sqrt{c^{2} - 20}}{2}$

Step 5.5

Factor $- 1$ out of $- \sqrt{c^{2} - 20}$ .

$x = \frac{- (c) - (\sqrt{c^{2} - 20})}{2}$

Step 5.6

Factor $- 1$ out of $- (c) - (\sqrt{c^{2} - 20})$ .

$x = \frac{- (c + \sqrt{c^{2} - 20})}{2}$

Step 5.7

Rewrite $- (c + \sqrt{c^{2} - 20})$ as $- 1 (c + \sqrt{c^{2} - 20})$ .

$x = \frac{- 1 (c + \sqrt{c^{2} - 20})}{2}$

Step 5.8

Move the negative in front of the fraction.

$x = - \frac{c + \sqrt{c^{2} - 20}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{c - \sqrt{c^{2} - 20}}{2}$

$x = - \frac{c + \sqrt{c^{2} - 20}}{2}$

Step 7

The given equation $x = - \frac{c - \sqrt{c^{2} - 20}}{2}, - \frac{c + \sqrt{c^{2} - 20}}{2}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$