Precalculus Examples

$f (x) = x^{2} + 14 x + 49$

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 $x$ .

$0 = x^{2} + 14 x + 49$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x^{2} + 14 x + 49 = 0$ .

$x^{2} + 14 x + 49 = 0$

Step 1.2.2

Factor using the perfect square rule.

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Step 1.2.2.1

Rewrite $49$ as $7^{2}$ .

$x^{2} + 14 x + 7^{2} = 0$

Step 1.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 1.2.2.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 7 + 7^{2} = 0$

Step 1.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 7$ .

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

Step 1.2.3

Set the $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 1.2.4

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 7, 0)$

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 $y$ .

$y = {(0)}^{2} + 14 (0) + 49$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0^{2} + 14 (0) + 49$

Step 2.2.2

Remove parentheses.

$y = {(0)}^{2} + 14 (0) + 49$

Step 2.2.3

Simplify ${(0)}^{2} + 14 (0) + 49$ .

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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 $0$ .

$y = 0 + 14 (0) + 49$

Step 2.2.3.1.2

Multiply $14$ by $0$ .

$y = 0 + 0 + 49$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 49$

Step 2.2.3.2.2

Add $0$ and $49$ .

$y = 49$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 49)$

Step 3

List the intersections.

x-intercept(s): $(- 7, 0)$

y-intercept(s): $(0, 49)$

Step 4