Basic Math Examples

$133 = 2 y^{2} + 3 y + 1$

Step 1

Rewrite the equation as $2 y^{2} + 3 y + 1 = 133$ .

$2 y^{2} + 3 y + 1 = 133$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $133$ from both sides of the equation.

$2 y^{2} + 3 y + 1 - 133 = 0$

Step 2.2

Subtract $133$ from $1$ .

$2 y^{2} + 3 y - 132 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = 3$ , and $c = - 132$ into the quadratic formula and solve for $y$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (2 \cdot - 132)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$y = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 132}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 132$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$y = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 132}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 132$ .

$y = \frac{- 3 \pm \sqrt{9 + 1056}}{2 \cdot 2}$

Step 5.1.3

Add $9$ and $1056$ .

$y = \frac{- 3 \pm \sqrt{1065}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$y = \frac{- 3 \pm \sqrt{1065}}{4}$

Step 6

The final answer is the combination of both solutions.

$y = - \frac{3 - \sqrt{1065}}{4}, - \frac{3 + \sqrt{1065}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{3 - \sqrt{1065}}{4}, - \frac{3 + \sqrt{1065}}{4}$

Decimal Form:

$y = 7.40858443 \dots, - 8.90858443 \dots$