Algebra Examples

$1$ , $2 - \sqrt{3}$

Step 1

$x = 1$ and $x = 2 - \sqrt{3}$ are the two real distinct solutions for the quadratic equation, which means that $x - 1$ and $x - 2 - \sqrt{3}$ are the factors of the quadratic equation.

$(x - 1) (x - 2 - \sqrt{3}) = 0$

Step 2

Expand $(x - 1) (x - 2 - \sqrt{3})$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x + x \cdot - 2 + x (- \sqrt{3}) - 1 x - 1 \cdot - 2 - 1 (- \sqrt{3}) = 0$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $- 2$ to the left of $x$ .

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

Step 3.1.3

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

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

Step 3.1.4

Multiply $- 1$ by $- 2$ .

$x^{2} - 2 x + x (- \sqrt{3}) - x + 2 - 1 (- \sqrt{3}) = 0$

Step 3.1.5

Multiply $- 1 (- \sqrt{3})$ .

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Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

$x^{2} - 2 x + x (- \sqrt{3}) - x + 2 + 1 \sqrt{3} = 0$

Step 3.1.5.2

Multiply $\sqrt{3}$ by $1$ .

$x^{2} - 2 x + x (- \sqrt{3}) - x + 2 + \sqrt{3} = 0$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Subtract $x$ from $- 2 x$ .

$x^{2} + x (- \sqrt{3}) - 3 x + 2 + \sqrt{3} = 0$

Step 3.2.2

Reorder factors in $x^{2} + x (- \sqrt{3}) - 3 x + 2 + \sqrt{3}$ .

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

Step 4

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

$y = x^{2} - x \sqrt{3} - 3 x + 2 + \sqrt{3}$

Step 5