Basic Math Examples

$- b^{4} + 3 b^{2} + \sqrt{10} = 0$

Step 1

Substitute $u = b^{2}$ into the equation. This will make the quadratic formula easy to use.

$- u^{2} + 3 u + \sqrt{10} = 0$

$u = b^{2}$

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 $a = - 1$ , $b = 3$ , and $c = \sqrt{10}$ into the quadratic formula and solve for $u$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 1 \cdot \sqrt{10})}}{2 \cdot - 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$u = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1 \cdot \sqrt{10}}}{2 \cdot - 1}$

Step 4.1.2

Multiply $- 4$ by $- 1$ .

$u = \frac{- 3 \pm \sqrt{9 + 4 \sqrt{10}}}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$u = \frac{- 3 \pm \sqrt{9 + 4 \sqrt{10}}}{- 2}$

Step 4.3

Simplify $\frac{- 3 \pm \sqrt{9 + 4 \sqrt{10}}}{- 2}$ .

$u = \frac{3 \pm \sqrt{9 + 4 \sqrt{10}}}{2}$

Step 5

The final answer is the combination of both solutions.

$u = \frac{3 + \sqrt{9 + 4 \sqrt{10}}}{2}, \frac{3 - \sqrt{9 + 4 \sqrt{10}}}{2}$

Step 6

Substitute the real value of $u = b^{2}$ back into the solved equation.

$b^{2} = 3.82643023$

${(b^{2})}^{1} = - 0.82643023$

Step 7

Solve the first equation for $b$ .

$b^{2} = 3.82643023$

Step 8

Solve the equation for $b$ .

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Step 8.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$b = \pm \sqrt{3.82643023}$

Step 8.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 8.2.1

First, use the positive value of the $\pm$ to find the first solution.

$b = \sqrt{3.82643023}$

Step 8.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$b = - \sqrt{3.82643023}$

Step 8.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$b = \sqrt{3.82643023}, - \sqrt{3.82643023}$

Step 9

Solve the second equation for $b$ .

${(b^{2})}^{1} = - 0.82643023$

Step 10

Solve the equation for $b$ .

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Step 10.1

Remove parentheses.

$b^{2} = - 0.82643023$

Step 10.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$b = \pm \sqrt{- 0.82643023}$

Step 10.3

Simplify $\pm \sqrt{- 0.82643023}$ .

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Step 10.3.1

Rewrite $- 0.82643023$ as $- 1 (0.82643023)$ .

$b = \pm \sqrt{- 1 (0.82643023)}$

Step 10.3.2

Rewrite $\sqrt{- 1 (0.82643023)}$ as $\sqrt{- 1} \cdot \sqrt{0.82643023}$ .

$b = \pm \sqrt{- 1} \cdot \sqrt{0.82643023}$

Step 10.3.3

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

$b = \pm i \sqrt{0.82643023}$

Step 10.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 10.4.1

First, use the positive value of the $\pm$ to find the first solution.

$b = i \sqrt{0.82643023}$

Step 10.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$b = - i \sqrt{0.82643023}$

Step 10.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$b = i \sqrt{0.82643023}, - i \sqrt{0.82643023}$

Step 11

The solution to $- b^{4} + 3 b^{2} + \sqrt{10} = 0$ is $b = \sqrt{3.82643023}, - \sqrt{3.82643023}, i \sqrt{0.82643023}, - i \sqrt{0.82643023}$ .

$b = \sqrt{3.82643023}, - \sqrt{3.82643023}, i \sqrt{0.82643023}, - i \sqrt{0.82643023}$