Precalculus Examples

$(1, 3)$

Step 1

$x = 1$ and $x = 3$ are the two real distinct solutions for the quadratic equation, which means that $x - 1$ and $x - 3$ are the factors of the quadratic equation.

$(x - 1) (x - 3) = 0$

Step 2

Expand $(x - 1) (x - 3)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - 3) - 1 (x - 3) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 1 (x - 3) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x \cdot - 3 - 1 x - 1 \cdot - 3 = 0$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 - 1 x - 1 \cdot - 3 = 0$

Step 3.1.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 \cdot x - 1 x - 1 \cdot - 3 = 0$

Step 3.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 3 x - x - 1 \cdot - 3 = 0$

Step 3.1.4

Multiply $- 1$ by $- 3$ .

$x^{2} - 3 x - x + 3 = 0$

Step 3.2

Subtract $x$ from $- 3 x$ .

$x^{2} - 4 x + 3 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${1, 3}$ is $y = x^{2} - 4 x + 3$ .

$y = x^{2} - 4 x + 3$

Step 5

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