Precalculus Examples

$y = 16 x^{2} - 8 x + 1$

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 $x$ .

$0 = 16 x^{2} - 8 x + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $16 x^{2} - 8 x + 1 = 0$ .

$16 x^{2} - 8 x + 1 = 0$

Step 1.2.2

Factor using the perfect square rule.

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Step 1.2.2.1

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

${(4 x)}^{2} - 8 x + 1 = 0$

Step 1.2.2.2

Rewrite $1$ as $1^{2}$ .

${(4 x)}^{2} - 8 x + 1^{2} = 0$

Step 1.2.2.3

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

$8 x = 2 \cdot (4 x) \cdot 1$

Step 1.2.2.4

Rewrite the polynomial.

${(4 x)}^{2} - 2 \cdot (4 x) \cdot 1 + 1^{2} = 0$

Step 1.2.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 4 x$ and $b = 1$ .

${(4 x - 1)}^{2} = 0$

Step 1.2.3

Set the $4 x - 1$ equal to $0$ .

$4 x - 1 = 0$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Add $1$ to both sides of the equation.

$4 x = 1$

Step 1.2.4.2

Divide each term in $4 x = 1$ by $4$ and simplify.

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Step 1.2.4.2.1

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $4$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Step 1.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{1}{4}, 0)$

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 $y$ .

$y = 16 {(0)}^{2} - 8 \cdot 0 + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 16 \cdot 0^{2} - 8 \cdot 0 + 1$

Step 2.2.2

Remove parentheses.

$y = 16 {(0)}^{2} - 8 \cdot 0 + 1$

Step 2.2.3

Simplify $16 {(0)}^{2} - 8 \cdot 0 + 1$ .

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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 $0$ .

$y = 16 \cdot 0 - 8 \cdot 0 + 1$

Step 2.2.3.1.2

Multiply $16$ by $0$ .

$y = 0 - 8 \cdot 0 + 1$

Step 2.2.3.1.3

Multiply $- 8$ by $0$ .

$y = 0 + 0 + 1$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 1$

Step 2.2.3.2.2

Add $0$ and $1$ .

$y = 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1)$

Step 3

List the intersections.

x-intercept(s): $(\frac{1}{4}, 0)$

y-intercept(s): $(0, 1)$

Step 4