Precalculus Examples

$(3, \frac{π}{6})$

Step 1

$x = 3$ and $x = \frac{π}{6}$ are the two real distinct solutions for the quadratic equation, which means that $x - 3$ and $x - \frac{π}{6}$ are the factors of the quadratic equation.

$(x - 3) (x - \frac{π}{6}) = 0$

Step 2

Expand $(x - 3) (x - \frac{π}{6})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - \frac{π}{6}) - 3 (x - \frac{π}{6}) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x (- \frac{π}{6}) - 3 (x - \frac{π}{6}) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x (- \frac{π}{6}) - 3 x - 3 (- \frac{π}{6}) = 0$

Step 3

Simplify each term.

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Step 3.1

Multiply $x$ by $x$ .

$x^{2} + x (- \frac{π}{6}) - 3 x - 3 (- \frac{π}{6}) = 0$

Step 3.2

Combine $x$ and $\frac{π}{6}$ .

$x^{2} - \frac{x π}{6} - 3 x - 3 (- \frac{π}{6}) = 0$

Step 3.3

Cancel the common factor of $3$ .

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Step 3.3.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$x^{2} - \frac{x π}{6} - 3 x - 3 \frac{- π}{6} = 0$

Step 3.3.2

Factor $3$ out of $- 3$ .

$x^{2} - \frac{x π}{6} - 3 x + 3 (- 1) (\frac{- π}{6}) = 0$

Step 3.3.3

Factor $3$ out of $6$ .

$x^{2} - \frac{x π}{6} - 3 x + 3 \cdot (- 1 \frac{- π}{3 \cdot 2}) = 0$

Step 3.3.4

Cancel the common factor.

$x^{2} - \frac{x π}{6} - 3 x + 3 \cdot (- 1 \frac{- π}{3 \cdot 2}) = 0$

Step 3.3.5

Rewrite the expression.

$x^{2} - \frac{x π}{6} - 3 x - 1 \frac{- π}{2} = 0$

Step 3.4

Move the negative in front of the fraction.

$x^{2} - \frac{x π}{6} - 3 x - 1 (- \frac{π}{2}) = 0$

Step 3.5

Multiply $- 1 (- \frac{π}{2})$ .

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Step 3.5.1

Multiply $- 1$ by $- 1$ .

$x^{2} - \frac{x π}{6} - 3 x + 1 (\frac{π}{2}) = 0$

Step 3.5.2

Multiply $\frac{π}{2}$ by $1$ .

$x^{2} - \frac{x π}{6} - 3 x + \frac{π}{2} = 0$

Step 4

The standard quadratic equation using the given set of solutions ${3, \frac{π}{6}}$ is $y = x^{2} - \frac{x π}{6} - 3 x + \frac{π}{2}$ .

$y = x^{2} - \frac{x π}{6} - 3 x + \frac{π}{2}$

Step 5