Precalculus Examples

$f (x) = - 2 x^{2} + 11 x - 14$

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 $x$ .

$0 = - 2 x^{2} + 11 x - 14$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 2 x^{2} + 11 x - 14 = 0$ .

$- 2 x^{2} + 11 x - 14 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 1$ out of $- 2 x^{2} + 11 x - 14$ .

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Step 1.2.2.1.1

Factor $- 1$ out of $- 2 x^{2}$ .

$- (2 x^{2}) + 11 x - 14 = 0$

Step 1.2.2.1.2

Factor $- 1$ out of $11 x$ .

$- (2 x^{2}) - (- 11 x) - 14 = 0$

Step 1.2.2.1.3

Rewrite $- 14$ as $- 1 (14)$ .

$- (2 x^{2}) - (- 11 x) - 1 \cdot 14 = 0$

Step 1.2.2.1.4

Factor $- 1$ out of $- (2 x^{2}) - (- 11 x)$ .

$- (2 x^{2} - 11 x) - 1 \cdot 14 = 0$

Step 1.2.2.1.5

Factor $- 1$ out of $- (2 x^{2} - 11 x) - 1 (14)$ .

$- (2 x^{2} - 11 x + 14) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor by grouping.

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Step 1.2.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 14 = 28$ and whose sum is $b = - 11$ .

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Step 1.2.2.2.1.1.1

Factor $- 11$ out of $- 11 x$ .

$- (2 x^{2} - 11 x + 14) = 0$

Step 1.2.2.2.1.1.2

Rewrite $- 11$ as $- 4$ plus $- 7$

$- (2 x^{2} + (- 4 - 7) x + 14) = 0$

Step 1.2.2.2.1.1.3

Apply the distributive property.

$- (2 x^{2} - 4 x - 7 x + 14) = 0$

Step 1.2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((2 x^{2} - 4 x) - 7 x + 14) = 0$

Step 1.2.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (2 x (x - 2) - 7 (x - 2)) = 0$

Step 1.2.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 2$ .

$- ((x - 2) (2 x - 7)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$- (x - 2) (2 x - 7) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$2 x - 7 = 0$

Step 1.2.4

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5

Set $2 x - 7$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 x - 7$ equal to $0$ .

$2 x - 7 = 0$

Step 1.2.5.2

Solve $2 x - 7 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $7$ to both sides of the equation.

$2 x = 7$

Step 1.2.5.2.2

Divide each term in $2 x = 7$ by $2$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $2 x = 7$ by $2$ .

$\frac{2 x}{2} = \frac{7}{2}$

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{7}{2}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{7}{2}$

Step 1.2.6

The final solution is all the values that make $- (x - 2) (2 x - 7) = 0$ true.

$x = 2, \frac{7}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (\frac{7}{2}, 0)$

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 $y$ .

$y = - 2 {(0)}^{2} + 11 (0) - 14$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 2 \cdot 0^{2} + 11 (0) - 14$

Step 2.2.2

Remove parentheses.

$y = - 2 {(0)}^{2} + 11 (0) - 14$

Step 2.2.3

Simplify $- 2 {(0)}^{2} + 11 (0) - 14$ .

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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 $0$ .

$y = - 2 \cdot 0 + 11 (0) - 14$

Step 2.2.3.1.2

Multiply $- 2$ by $0$ .

$y = 0 + 11 (0) - 14$

Step 2.2.3.1.3

Multiply $11$ by $0$ .

$y = 0 + 0 - 14$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 14$

Step 2.2.3.2.2

Subtract $14$ from $0$ .

$y = - 14$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 14)$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (\frac{7}{2}, 0)$

y-intercept(s): $(0, - 14)$

Step 4