Precalculus Examples

$f (x) = - 7 x^{2} + 6 x + 2$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 7 x^{2} + 6 x + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 7 x^{2} + 6 x + 2 = 0$ .

$- 7 x^{2} + 6 x + 2 = 0$

Step 1.2.2

Use the quadratic formula to find the solutions.

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

Step 1.2.3

Substitute the values $a = - 7$ , $b = 6$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 7 \cdot 2)}}{2 \cdot - 7}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 7 \cdot 2}}{2 \cdot - 7}$

Step 1.2.4.1.2

Multiply $- 4 \cdot - 7 \cdot 2$ .

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Step 1.2.4.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 + 28 \cdot 2}}{2 \cdot - 7}$

Step 1.2.4.1.2.2

Multiply $28$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 + 56}}{2 \cdot - 7}$

Step 1.2.4.1.3

Add $36$ and $56$ .

$x = \frac{- 6 \pm \sqrt{92}}{2 \cdot - 7}$

Step 1.2.4.1.4

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 1.2.4.1.4.1

Factor $4$ out of $92$ .

$x = \frac{- 6 \pm \sqrt{4 (23)}}{2 \cdot - 7}$

Step 1.2.4.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 23}}{2 \cdot - 7}$

Step 1.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{23}}{2 \cdot - 7}$

Step 1.2.4.2

Multiply $2$ by $- 7$ .

$x = \frac{- 6 \pm 2 \sqrt{23}}{- 14}$

Step 1.2.4.3

Simplify $\frac{- 6 \pm 2 \sqrt{23}}{- 14}$ .

$x = \frac{3 \pm \sqrt{23}}{7}$

Step 1.2.5

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

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 7 \cdot 2}}{2 \cdot - 7}$

Step 1.2.5.1.2

Multiply $- 4 \cdot - 7 \cdot 2$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 + 28 \cdot 2}}{2 \cdot - 7}$

Step 1.2.5.1.2.2

Multiply $28$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 + 56}}{2 \cdot - 7}$

Step 1.2.5.1.3

Add $36$ and $56$ .

$x = \frac{- 6 \pm \sqrt{92}}{2 \cdot - 7}$

Step 1.2.5.1.4

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 1.2.5.1.4.1

Factor $4$ out of $92$ .

$x = \frac{- 6 \pm \sqrt{4 (23)}}{2 \cdot - 7}$

Step 1.2.5.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 23}}{2 \cdot - 7}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{23}}{2 \cdot - 7}$

Step 1.2.5.2

Multiply $2$ by $- 7$ .

$x = \frac{- 6 \pm 2 \sqrt{23}}{- 14}$

Step 1.2.5.3

Simplify $\frac{- 6 \pm 2 \sqrt{23}}{- 14}$ .

$x = \frac{3 \pm \sqrt{23}}{7}$

Step 1.2.5.4

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{23}}{7}$

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 7 \cdot 2}}{2 \cdot - 7}$

Step 1.2.6.1.2

Multiply $- 4 \cdot - 7 \cdot 2$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $- 7$ .

$x = \frac{- 6 \pm \sqrt{36 + 28 \cdot 2}}{2 \cdot - 7}$

Step 1.2.6.1.2.2

Multiply $28$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 + 56}}{2 \cdot - 7}$

Step 1.2.6.1.3

Add $36$ and $56$ .

$x = \frac{- 6 \pm \sqrt{92}}{2 \cdot - 7}$

Step 1.2.6.1.4

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 1.2.6.1.4.1

Factor $4$ out of $92$ .

$x = \frac{- 6 \pm \sqrt{4 (23)}}{2 \cdot - 7}$

Step 1.2.6.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 23}}{2 \cdot - 7}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{23}}{2 \cdot - 7}$

Step 1.2.6.2

Multiply $2$ by $- 7$ .

$x = \frac{- 6 \pm 2 \sqrt{23}}{- 14}$

Step 1.2.6.3

Simplify $\frac{- 6 \pm 2 \sqrt{23}}{- 14}$ .

$x = \frac{3 \pm \sqrt{23}}{7}$

Step 1.2.6.4

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{23}}{7}$

Step 1.2.7

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{23}}{7}, \frac{3 - \sqrt{23}}{7}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3 + \sqrt{23}}{7}, 0), (\frac{3 - \sqrt{23}}{7}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 7 {(0)}^{2} + 6 (0) + 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 7 \cdot 0^{2} + 6 (0) + 2$

Step 2.2.2

Remove parentheses.

$y = - 7 {(0)}^{2} + 6 (0) + 2$

Step 2.2.3

Simplify $- 7 {(0)}^{2} + 6 (0) + 2$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = - 7 \cdot 0 + 6 (0) + 2$

Step 2.2.3.1.2

Multiply $- 7$ by $0$ .

$y = 0 + 6 (0) + 2$

Step 2.2.3.1.3

Multiply $6$ by $0$ .

$y = 0 + 0 + 2$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 2$

Step 2.2.3.2.2

Add $0$ and $2$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3 + \sqrt{23}}{7}, 0), (\frac{3 - \sqrt{23}}{7}, 0)$

y-intercept(s): $(0, 2)$

Step 4