Precalculus Examples

$x^{2} - 3 x - 7$

Step 1

Set $x^{2} - 3 x - 7$ equal to $0$ .

$x^{2} - 3 x - 7 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{3 \pm \sqrt{9 + 28}}{2 \cdot 1}$

Step 2.3.1.3

Add $9$ and $28$ .

$x = \frac{3 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{37}}{2}$

Step 2.4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{37}}{2}, \frac{3 - \sqrt{37}}{2}$

$x = \frac{3 \pm \sqrt{37}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 \pm \sqrt{37}}{2}$

Decimal Form:

$x = 4.54138126 \dots, - 1.54138126 \dots$

Step 4