Basic Math Examples

y^{2} - y = 3

$y^{2} - y = 3$

Step 1

Subtract

3

$3$ from both sides of the equation.

y^{2} - y - 3 = 0

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

Step 2

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 3

Substitute the values

a = 1

$a = 1$ ,

b = - 1

$b = - 1$ , and

c = - 3

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

y

$y$ .

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

y = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}

$y = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Step 4.1.2

Multiply

- 4 \cdot 1 \cdot - 3

$- 4 \cdot 1 \cdot - 3$ .

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Step 4.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

y = \frac{1 \pm \sqrt{1 - 4 \cdot - 3}}{2 \cdot 1}

$y = \frac{1 \pm \sqrt{1 - 4 \cdot - 3}}{2 \cdot 1}$

Step 4.1.2.2

Multiply

- 4

$- 4$ by

- 3

$- 3$ .

y = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 1}

$y = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 1}$

y = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 1}

$y = \frac{1 \pm \sqrt{1 + 12}}{2 \cdot 1}$

Step 4.1.3

Add

1

$1$ and

12

$12$ .

y = \frac{1 \pm \sqrt{13}}{2 \cdot 1}

$y = \frac{1 \pm \sqrt{13}}{2 \cdot 1}$

y = \frac{1 \pm \sqrt{13}}{2 \cdot 1}

$y = \frac{1 \pm \sqrt{13}}{2 \cdot 1}$

Step 4.2

Multiply

2

$2$ by

1

$1$ .

y = \frac{1 \pm \sqrt{13}}{2}

$y = \frac{1 \pm \sqrt{13}}{2}$

y = \frac{1 \pm \sqrt{13}}{2}

$y = \frac{1 \pm \sqrt{13}}{2}$

Step 5

The final answer is the combination of both solutions.

y = \frac{1 + \sqrt{13}}{2}, \frac{1 - \sqrt{13}}{2}

$y = \frac{1 + \sqrt{13}}{2}, \frac{1 - \sqrt{13}}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

y = \frac{1 + \sqrt{13}}{2}, \frac{1 - \sqrt{13}}{2}

$y = \frac{1 + \sqrt{13}}{2}, \frac{1 - \sqrt{13}}{2}$

Decimal Form:

y = 2.30277563 \dots, - 1.30277563 \dots

$y = 2.30277563 \dots, - 1.30277563 \dots$