Algebra Examples

$y = x^{2} - 3$ , $y = x + 1$

Step 1

Eliminate the equal sides of each equation and combine.

$x^{2} - 3 = x + 1$

Step 2

Solve $x^{2} - 3 = x + 1$ for $x$ .

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Step 2.1

Subtract $x$ from both sides of the equation.

$x^{2} - 3 - x = 1$

Step 2.2

Move all terms to the left side of the equation and simplify.

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Step 2.2.1

Subtract $1$ from both sides of the equation.

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

Step 2.2.2

Subtract $1$ from $- 3$ .

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

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = x + 1$

Step 2.4

Substitute the values $a = 1$ , $b = - 1$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1} + y = x + 1$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.5.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

Step 2.5.1.3

Add $1$ and $16$ .

$x = \frac{1 \pm \sqrt{17}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{17}}{2}$

Step 2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.6.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

Step 2.6.1.3

Add $1$ and $16$ .

$x = \frac{1 \pm \sqrt{17}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{17}}{2}$

Step 2.6.3

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{17}}{2}$

Step 2.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.7.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{1 \pm \sqrt{1 + 16}}{2 \cdot 1}$

Step 2.7.1.3

Add $1$ and $16$ .

$x = \frac{1 \pm \sqrt{17}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{17}}{2}$

Step 2.7.3

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{17}}{2}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{17}}{2}, \frac{1 - \sqrt{17}}{2}$

Step 3

Evaluate $y$ when $x = \frac{1 + \sqrt{17}}{2}$ .

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Step 3.1

Substitute $\frac{1 + \sqrt{17}}{2}$ for $x$ .

$y = (\frac{1 + \sqrt{17}}{2}) + 1$

Step 3.2

Substitute $\frac{1 + \sqrt{17}}{2}$ for $x$ in $y = (\frac{1 + \sqrt{17}}{2}) + 1$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = \frac{1 + \sqrt{17}}{2} + 1$

Step 3.2.2

Remove parentheses.

$y = (\frac{1 + \sqrt{17}}{2}) + 1$

Step 3.2.3

Simplify $(\frac{1 + \sqrt{17}}{2}) + 1$ .

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Step 3.2.3.1

Write $1$ as a fraction with a common denominator.

$y = \frac{1 + \sqrt{17}}{2} + \frac{2}{2}$

Step 3.2.3.2

Combine the numerators over the common denominator.

$y = \frac{1 + \sqrt{17} + 2}{2}$

Step 3.2.3.3

Add $1$ and $2$ .

$y = \frac{3 + \sqrt{17}}{2}$

Step 4

Evaluate $y$ when $x = \frac{1 - \sqrt{17}}{2}$ .

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Step 4.1

Substitute $\frac{1 - \sqrt{17}}{2}$ for $x$ .

$y = (\frac{1 - \sqrt{17}}{2}) + 1$

Step 4.2

Substitute $\frac{1 - \sqrt{17}}{2}$ for $x$ in $y = (\frac{1 - \sqrt{17}}{2}) + 1$ and solve for $y$ .

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Step 4.2.1

Remove parentheses.

$y = \frac{1 - \sqrt{17}}{2} + 1$

Step 4.2.2

Remove parentheses.

$y = (\frac{1 - \sqrt{17}}{2}) + 1$

Step 4.2.3

Simplify $(\frac{1 - \sqrt{17}}{2}) + 1$ .

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Step 4.2.3.1

Write $1$ as a fraction with a common denominator.

$y = \frac{1 - \sqrt{17}}{2} + \frac{2}{2}$

Step 4.2.3.2

Combine the numerators over the common denominator.

$y = \frac{1 - \sqrt{17} + 2}{2}$

Step 4.2.3.3

Add $1$ and $2$ .

$y = \frac{3 - \sqrt{17}}{2}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(\frac{1 + \sqrt{17}}{2}, \frac{3 + \sqrt{17}}{2})$

$(\frac{1 - \sqrt{17}}{2}, \frac{3 - \sqrt{17}}{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{1 + \sqrt{17}}{2}, \frac{3 + \sqrt{17}}{2}), (\frac{1 - \sqrt{17}}{2}, \frac{3 - \sqrt{17}}{2})$

Equation Form:

$x = \frac{1 + \sqrt{17}}{2}, y = \frac{3 + \sqrt{17}}{2}$

$x = \frac{1 - \sqrt{17}}{2}, y = \frac{3 - \sqrt{17}}{2}$

Step 7