Finite Math Examples

$0 = x^{2} + 4 x - 10$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $x^{2}$ from both sides of the equation.

$- x^{2} = 4 x - 10$

Step 1.2

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

$- x^{2} - 4 x = - 10$

Step 1.3

Add $10$ to both sides of the equation.

$- x^{2} - 4 x + 10 = 0$

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 $a = - 1$ , $b = - 4$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot 10)}}{2 \cdot - 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 10}}{2 \cdot - 1}$

Step 4.1.2

Multiply $- 4 \cdot - 1 \cdot 10$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot 10}}{2 \cdot - 1}$

Step 4.1.2.2

Multiply $4$ by $10$ .

$x = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot - 1}$

Step 4.1.3

Add $16$ and $40$ .

$x = \frac{4 \pm \sqrt{56}}{2 \cdot - 1}$

Step 4.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 4.1.4.1

Factor $4$ out of $56$ .

$x = \frac{4 \pm \sqrt{4 (14)}}{2 \cdot - 1}$

Step 4.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot - 1}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{14}}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 2 \sqrt{14}}{- 2}$

Step 4.3

Simplify $\frac{4 \pm 2 \sqrt{14}}{- 2}$ .

$x = \frac{2 \pm \sqrt{14}}{- 1}$

Step 4.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{14}}{- 1}$ .

$x = - 1 \cdot (2 \pm \sqrt{14})$

Step 4.5

Rewrite $- 1 \cdot (2 \pm \sqrt{14})$ as $- (2 \pm \sqrt{14})$ .

$x = - (2 \pm \sqrt{14})$

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 10}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot 10$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot 10}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $10$ .

$x = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot - 1}$

Step 5.1.3

Add $16$ and $40$ .

$x = \frac{4 \pm \sqrt{56}}{2 \cdot - 1}$

Step 5.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 5.1.4.1

Factor $4$ out of $56$ .

$x = \frac{4 \pm \sqrt{4 (14)}}{2 \cdot - 1}$

Step 5.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot - 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{14}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 2 \sqrt{14}}{- 2}$

Step 5.3

Simplify $\frac{4 \pm 2 \sqrt{14}}{- 2}$ .

$x = \frac{2 \pm \sqrt{14}}{- 1}$

Step 5.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{14}}{- 1}$ .

$x = - 1 \cdot (2 \pm \sqrt{14})$

Step 5.5

Rewrite $- 1 \cdot (2 \pm \sqrt{14})$ as $- (2 \pm \sqrt{14})$ .

$x = - (2 \pm \sqrt{14})$

Step 5.6

Change the $\pm$ to $+$ .

$x = - (2 + \sqrt{14})$

Step 5.7

Apply the distributive property.

$x = - 1 \cdot 2 - \sqrt{14}$

Step 5.8

Multiply $- 1$ by $2$ .

$x = - 2 - \sqrt{14}$

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 10}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot 10$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot 10}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $10$ .

$x = \frac{4 \pm \sqrt{16 + 40}}{2 \cdot - 1}$

Step 6.1.3

Add $16$ and $40$ .

$x = \frac{4 \pm \sqrt{56}}{2 \cdot - 1}$

Step 6.1.4

Rewrite $56$ as $2^{2} \cdot 14$ .

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Step 6.1.4.1

Factor $4$ out of $56$ .

$x = \frac{4 \pm \sqrt{4 (14)}}{2 \cdot - 1}$

Step 6.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot - 1}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{14}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$x = \frac{4 \pm 2 \sqrt{14}}{- 2}$

Step 6.3

Simplify $\frac{4 \pm 2 \sqrt{14}}{- 2}$ .

$x = \frac{2 \pm \sqrt{14}}{- 1}$

Step 6.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{14}}{- 1}$ .

$x = - 1 \cdot (2 \pm \sqrt{14})$

Step 6.5

Rewrite $- 1 \cdot (2 \pm \sqrt{14})$ as $- (2 \pm \sqrt{14})$ .

$x = - (2 \pm \sqrt{14})$

Step 6.6

Change the $\pm$ to $-$ .

$x = - (2 - \sqrt{14})$

Step 6.7

Apply the distributive property.

$x = - 1 \cdot 2 + \sqrt{14}$

Step 6.8

Multiply $- 1$ by $2$ .

$x = - 2 + \sqrt{14}$

Step 6.9

Multiply $- - \sqrt{14}$ .

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Step 6.9.1

Multiply $- 1$ by $- 1$ .

$x = - 2 + 1 \sqrt{14}$

Step 6.9.2

Multiply $\sqrt{14}$ by $1$ .

$x = - 2 + \sqrt{14}$

Step 7

The final answer is the combination of both solutions.

$x = - 2 - \sqrt{14}, - 2 + \sqrt{14}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - 2 - \sqrt{14}, - 2 + \sqrt{14}$

Decimal Form:

$x = - 5.74165738 \dots, 1.74165738 \dots$