Examples

$(1, 5)$

Step 1

$x = 1$ and $x = 5$ are the two real distinct solutions for the quadratic equation, which means that $x - 1$ and $x - 5$ are the factors of the quadratic equation.

$(x - 1) (x - 5) = 0$

Step 2

Expand $(x - 1) (x - 5)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x (x - 5) - 1 (x - 5) = 0$

Step 2.2

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 1 (x - 5) = 0$

Step 2.3

Apply the distributive property.

$x \cdot x + x \cdot - 5 - 1 x - 1 \cdot - 5 = 0$

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 $x$ .

$x^{2} + x \cdot - 5 - 1 x - 1 \cdot - 5 = 0$

Step 3.1.2

Move $- 5$ to the left of $x$ .

$x^{2} - 5 \cdot x - 1 x - 1 \cdot - 5 = 0$

Step 3.1.3

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

$x^{2} - 5 x - x - 1 \cdot - 5 = 0$

Step 3.1.4

Multiply $- 1$ by $- 5$ .

$x^{2} - 5 x - x + 5 = 0$

Step 3.2

Subtract $x$ from $- 5 x$ .

$x^{2} - 6 x + 5 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${1, 5}$ is $y = x^{2} - 6 x + 5$ .

$y = x^{2} - 6 x + 5$

Step 5

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