Finite Math Examples

$y = 0.5 x^{2} + x$

Step 1

Rewrite the equation as $0.5 x^{2} + x = y$ .

$0.5 x^{2} + x = y$

Step 2

Subtract $y$ from both sides of the equation.

$0.5 x^{2} + x - y = 0$

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 $a = 0.5$ , $b = 1$ , and $c = - y$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (0.5 \cdot (- y))}}{2 \cdot 0.5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 0.5 \cdot (- 1 y)}}{2 \cdot 0.5}$

Step 5.1.2

Multiply $- 4 \cdot 0.5 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $0.5$ .

$x = \frac{- 1 \pm \sqrt{1 - 2 \cdot (- 1 y)}}{2 \cdot 0.5}$

Step 5.1.2.2

Multiply $- 2$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 2 y}}{2 \cdot 0.5}$

Step 5.2

Multiply $2$ by $0.5$ .

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

Step 5.3

Divide $- 1 \pm \sqrt{1 + 2 y}$ by $1$ .

$x = - 1 \pm \sqrt{1 + 2 y}$

Step 6

The final answer is the combination of both solutions.

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

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