Pre-Algebra Examples

$\sqrt{9 x^{2} - 65 x + 14}$

Step 1

Find the $y$ value at $x = 0$ .

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Step 1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \sqrt{(9 (0) - 2) ((0) - 7)}$

Step 1.2

Simplify the result.

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Step 1.2.1

Multiply $9$ by $0$ .

$f (0) = \sqrt{(0 - 2) (0 - 7)}$

Step 1.2.2

Subtract $2$ from $0$ .

$f (0) = \sqrt{- 2 (0 - 7)}$

Step 1.2.3

Subtract $7$ from $0$ .

$f (0) = \sqrt{- 2 \cdot - 7}$

Step 1.2.4

Multiply $- 2$ by $- 7$ .

$f (0) = \sqrt{14}$

Step 1.2.5

The final answer is $\sqrt{14}$ .

$\sqrt{14}$

Step 1.3

The $y$ value at $x = 0$ is $\sqrt{14}$ .

$y = \sqrt{14}$

Step 2

Find the $y$ value at $x = - 1$ .

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Step 2.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \sqrt{(9 (- 1) - 2) ((- 1) - 7)}$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $9$ by $- 1$ .

$f (- 1) = \sqrt{(- 9 - 2) (- 1 - 7)}$

Step 2.2.2

Subtract $2$ from $- 9$ .

$f (- 1) = \sqrt{- 11 (- 1 - 7)}$

Step 2.2.3

Subtract $7$ from $- 1$ .

$f (- 1) = \sqrt{- 11 \cdot - 8}$

Step 2.2.4

Multiply $- 11$ by $- 8$ .

$f (- 1) = \sqrt{88}$

Step 2.2.5

Rewrite $88$ as $2^{2} \cdot 22$ .

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Step 2.2.5.1

Factor $4$ out of $88$ .

$f (- 1) = \sqrt{4 (22)}$

Step 2.2.5.2

Rewrite $4$ as $2^{2}$ .

$f (- 1) = \sqrt{2^{2} \cdot 22}$

Step 2.2.6

Pull terms out from under the radical.

$f (- 1) = 2 \sqrt{22}$

Step 2.2.7

The final answer is $2 \sqrt{22}$ .

$2 \sqrt{22}$

Step 2.3

The $y$ value at $x = - 1$ is $2 \sqrt{22}$ .

$y = 2 \sqrt{22}$

Step 3

Find the $y$ value at $x = - 2$ .

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Step 3.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \sqrt{(9 (- 2) - 2) ((- 2) - 7)}$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $9$ by $- 2$ .

$f (- 2) = \sqrt{(- 18 - 2) (- 2 - 7)}$

Step 3.2.2

Subtract $2$ from $- 18$ .

$f (- 2) = \sqrt{- 20 (- 2 - 7)}$

Step 3.2.3

Subtract $7$ from $- 2$ .

$f (- 2) = \sqrt{- 20 \cdot - 9}$

Step 3.2.4

Multiply $- 20$ by $- 9$ .

$f (- 2) = \sqrt{180}$

Step 3.2.5

Rewrite $180$ as $6^{2} \cdot 5$ .

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Step 3.2.5.1

Factor $36$ out of $180$ .

$f (- 2) = \sqrt{36 (5)}$

Step 3.2.5.2

Rewrite $36$ as $6^{2}$ .

$f (- 2) = \sqrt{6^{2} \cdot 5}$

Step 3.2.6

Pull terms out from under the radical.

$f (- 2) = 6 \sqrt{5}$

Step 3.2.7

The final answer is $6 \sqrt{5}$ .

$6 \sqrt{5}$

Step 3.3

The $y$ value at $x = - 2$ is $6 \sqrt{5}$ .

$y = 6 \sqrt{5}$

Step 4

Find the $y$ value at $x = 7$ .

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Step 4.1

Replace the variable $x$ with $7$ in the expression.

$f (7) = \sqrt{(9 (7) - 2) ((7) - 7)}$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $9$ by $7$ .

$f (7) = \sqrt{(63 - 2) (7 - 7)}$

Step 4.2.2

Subtract $2$ from $63$ .

$f (7) = \sqrt{61 (7 - 7)}$

Step 4.2.3

Subtract $7$ from $7$ .

$f (7) = \sqrt{61 \cdot 0}$

Step 4.2.4

Multiply $61$ by $0$ .

$f (7) = \sqrt{0}$

Step 4.2.5

Rewrite $0$ as $0^{2}$ .

$f (7) = \sqrt{0^{2}}$

Step 4.2.6

Pull terms out from under the radical, assuming positive real numbers.

$f (7) = 0$

Step 4.2.7

The final answer is $0$ .

$0$

Step 4.3

The $y$ value at $x = 7$ is $0$ .

$y = 0$

Step 5

Find the $y$ value at $x = 8$ .

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Step 5.1

Replace the variable $x$ with $8$ in the expression.

$f (8) = \sqrt{(9 (8) - 2) ((8) - 7)}$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $9$ by $8$ .

$f (8) = \sqrt{(72 - 2) (8 - 7)}$

Step 5.2.2

Subtract $2$ from $72$ .

$f (8) = \sqrt{70 (8 - 7)}$

Step 5.2.3

Subtract $7$ from $8$ .

$f (8) = \sqrt{70 \cdot 1}$

Step 5.2.4

Multiply $70$ by $1$ .

$f (8) = \sqrt{70}$

Step 5.2.5

The final answer is $\sqrt{70}$ .

$\sqrt{70}$

Step 5.3

The $y$ value at $x = 8$ is $\sqrt{70}$ .

$y = \sqrt{70}$

Step 6

Find the $y$ value at $x = 9$ .

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Step 6.1

Replace the variable $x$ with $9$ in the expression.

$f (9) = \sqrt{(9 (9) - 2) ((9) - 7)}$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $9$ by $9$ .

$f (9) = \sqrt{(81 - 2) (9 - 7)}$

Step 6.2.2

Subtract $2$ from $81$ .

$f (9) = \sqrt{79 (9 - 7)}$

Step 6.2.3

Subtract $7$ from $9$ .

$f (9) = \sqrt{79 \cdot 2}$

Step 6.2.4

Multiply $79$ by $2$ .

$f (9) = \sqrt{158}$

Step 6.2.5

The final answer is $\sqrt{158}$ .

$\sqrt{158}$

Step 6.3

The $y$ value at $x = 9$ is $\sqrt{158}$ .

$y = \sqrt{158}$

Step 7

List the points to graph.

$(0, 3.74165738), (- 1, 9.38083151), (- 2, 13.41640786), (7, 0), (8, 8.36660026), (9, 12.56980508)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 2 & 13.416 \\ - 1 & 9.381 \\ 0 & 3.742 \\ 7 & 0 \\ 8 & 8.367 \\ 9 & 12.57 \end{array}$

Step 9