Algebra Examples

${(0.8 x + 4.1)}^{2} - 1.3$

Step 1

Write ${(0.8 x + 4.1)}^{2} - 1.3$ as an equation.

$y = {(0.8 x + 4.1)}^{2} - 1.3$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = {(0.8 x + 4.1)}^{2} - 1.3$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as ${(0.8 x + 4.1)}^{2} - 1.3 = 0$ .

${(0.8 x + 4.1)}^{2} - 1.3 = 0$

Step 2.2.2

Add $1.3$ to both sides of the equation.

${(0.8 x + 4.1)}^{2} = 1.3$

Step 2.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$0.8 x + 4.1 = \pm \sqrt{1.3}$

Step 2.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$0.8 x + 4.1 = \sqrt{1.3}$

Step 2.2.4.2

Subtract $4.1$ from both sides of the equation.

$0.8 x = \sqrt{1.3} - 4.1$

Step 2.2.4.3

Divide each term in $0.8 x = \sqrt{1.3} - 4.1$ by $0.8$ and simplify.

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Step 2.2.4.3.1

Divide each term in $0.8 x = \sqrt{1.3} - 4.1$ by $0.8$ .

$\frac{0.8 x}{0.8} = \frac{\sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.2

Simplify the left side.

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Step 2.2.4.3.2.1

Cancel the common factor of $0.8$ .

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Step 2.2.4.3.2.1.1

Cancel the common factor.

$\frac{0.8 x}{0.8} = \frac{\sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.3

Simplify the right side.

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Step 2.2.4.3.3.1

Simplify each term.

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Step 2.2.4.3.3.1.1

Multiply by $1$ .

$x = \frac{1 \sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.3.1.2

Factor $0.8$ out of $0.8$ .

$x = \frac{1 \sqrt{1.3}}{0.8 (1)} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.3.1.3

Separate fractions.

$x = \frac{1}{0.8} \cdot \frac{\sqrt{1.3}}{1} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.3.1.4

Divide $1$ by $0.8$ .

$x = 1.25 \frac{\sqrt{1.3}}{1} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.3.1.5

Divide $\sqrt{1.3}$ by $1$ .

$x = 1.25 \sqrt{1.3} + \frac{- 4.1}{0.8}$

Step 2.2.4.3.3.1.6

Divide $- 4.1$ by $0.8$ .

$x = 1.25 \sqrt{1.3} - 5.125$

Step 2.2.4.4

Next, use the negative value of the $\pm$ to find the second solution.

$0.8 x + 4.1 = - \sqrt{1.3}$

Step 2.2.4.5

Subtract $4.1$ from both sides of the equation.

$0.8 x = - \sqrt{1.3} - 4.1$

Step 2.2.4.6

Divide each term in $0.8 x = - \sqrt{1.3} - 4.1$ by $0.8$ and simplify.

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Step 2.2.4.6.1

Divide each term in $0.8 x = - \sqrt{1.3} - 4.1$ by $0.8$ .

$\frac{0.8 x}{0.8} = \frac{- \sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.6.2

Simplify the left side.

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Step 2.2.4.6.2.1

Cancel the common factor of $0.8$ .

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Step 2.2.4.6.2.1.1

Cancel the common factor.

$\frac{0.8 x}{0.8} = \frac{- \sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3

Simplify the right side.

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Step 2.2.4.6.3.1

Simplify each term.

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Step 2.2.4.6.3.1.1

Move the negative in front of the fraction.

$x = - \frac{\sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3.1.2

Multiply by $1$ .

$x = - \frac{1 \sqrt{1.3}}{0.8} + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3.1.3

Factor $0.8$ out of $0.8$ .

$x = - \frac{1 \sqrt{1.3}}{0.8 (1)} + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3.1.4

Separate fractions.

$x = - (\frac{1}{0.8} \cdot \frac{\sqrt{1.3}}{1}) + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3.1.5

Divide $1$ by $0.8$ .

$x = - (1.25 \frac{\sqrt{1.3}}{1}) + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3.1.6

Divide $\sqrt{1.3}$ by $1$ .

$x = - (1.25 \sqrt{1.3}) + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3.1.7

Multiply $1.25$ by $- 1$ .

$x = - 1.25 \sqrt{1.3} + \frac{- 4.1}{0.8}$

Step 2.2.4.6.3.1.8

Divide $- 4.1$ by $0.8$ .

$x = - 1.25 \sqrt{1.3} - 5.125$

Step 2.2.4.7

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1.25 \sqrt{1.3} - 5.125, - 1.25 \sqrt{1.3} - 5.125$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(1.25 \sqrt{1.3} - 5.125, 0), (- 1.25 \sqrt{1.3} - 5.125, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = {(0.8 (0) + 4.1)}^{2} - 1.3$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = {(0.8 (0) + 4.1)}^{2} - 1.3$

Step 3.2.2

Simplify ${(0.8 (0) + 4.1)}^{2} - 1.3$ .

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $0.8$ by $0$ .

$y = {(0 + 4.1)}^{2} - 1.3$

Step 3.2.2.1.2

Add $0$ and $4.1$ .

$y = {4.1}^{2} - 1.3$

Step 3.2.2.1.3

Raise $4.1$ to the power of $2$ .

$y = 16.81 - 1.3$

Step 3.2.2.2

Subtract $1.3$ from $16.81$ .

$y = 15.51$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 15.51)$

Step 4

List the intersections.

x-intercept(s): $(1.25 \sqrt{1.3} - 5.125, 0), (- 1.25 \sqrt{1.3} - 5.125, 0)$

y-intercept(s): $(0, 15.51)$

Step 5