Basic Math Examples

12 p^{2} + b p + 3 = 0

$12 p^{2} + b p + 3 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

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

Step 2

Substitute the values

a = 12

$a = 12$ ,

b = b

$b = b$ , and

c = 3

$c = 3$ into the quadratic formula and solve for

p

$p$ .

\frac{- b \pm \sqrt{b^{2} - 4 \cdot (12 \cdot 3)}}{2 \cdot 12}

$\frac{- b \pm \sqrt{b^{2} - 4 \cdot (12 \cdot 3)}}{2 \cdot 12}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply

- 4 \cdot 12 \cdot 3

$- 4 \cdot 12 \cdot 3$ .

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Step 3.1.1.1

Multiply

- 4

$- 4$ by

12

$12$ .

p = \frac{- b \pm \sqrt{b^{2} - 48 \cdot 3}}{2 \cdot 12}

$p = \frac{- b \pm \sqrt{b^{2} - 48 \cdot 3}}{2 \cdot 12}$

Step 3.1.1.2

Multiply

- 48

$- 48$ by

3

$3$ .

p = \frac{- b \pm \sqrt{b^{2} - 144}}{2 \cdot 12}

$p = \frac{- b \pm \sqrt{b^{2} - 144}}{2 \cdot 12}$

p = \frac{- b \pm \sqrt{b^{2} - 144}}{2 \cdot 12}

$p = \frac{- b \pm \sqrt{b^{2} - 144}}{2 \cdot 12}$

Step 3.1.2

Rewrite

b^{2} - 144

$b^{2} - 144$ in a factored form.

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Step 3.1.2.1

Rewrite

144

$144$ as

12^{2}

$12^{2}$ .

p = \frac{- b \pm \sqrt{b^{2} - 12^{2}}}{2 \cdot 12}

$p = \frac{- b \pm \sqrt{b^{2} - 12^{2}}}{2 \cdot 12}$

Step 3.1.2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = b

$a = b$ and

b = 12

$b = 12$ .

p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{2 \cdot 12}

$p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{2 \cdot 12}$

p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{2 \cdot 12}

$p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{2 \cdot 12}$

p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{2 \cdot 12}

$p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{2 \cdot 12}$

Step 3.2

Multiply

2

$2$ by

12

$12$ .

p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{24}

$p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{24}$

p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{24}

$p = \frac{- b \pm \sqrt{(b + 12) (b - 12)}}{24}$

Step 4

The final answer is the combination of both solutions.

p = - \frac{b - \sqrt{(b + 12) (b - 12)}}{24}

$p = - \frac{b - \sqrt{(b + 12) (b - 12)}}{24}$

p = - \frac{b + \sqrt{(b + 12) (b - 12)}}{24}

$p = - \frac{b + \sqrt{(b + 12) (b - 12)}}{24}$