Algebra Examples

$y = 6 x^{2} - 7 x - 5$

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 $x$ .

$0 = 6 x^{2} - 7 x - 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $6 x^{2} - 7 x - 5 = 0$ .

$6 x^{2} - 7 x - 5 = 0$

Step 1.2.2

Factor by grouping.

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Step 1.2.2.1

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 $a \cdot c = 6 \cdot - 5 = - 30$ and whose sum is $b = - 7$ .

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Step 1.2.2.1.1

Factor $- 7$ out of $- 7 x$ .

$6 x^{2} - 7 x - 5 = 0$

Step 1.2.2.1.2

Rewrite $- 7$ as $3$ plus $- 10$

$6 x^{2} + (3 - 10) x - 5 = 0$

Step 1.2.2.1.3

Apply the distributive property.

$6 x^{2} + 3 x - 10 x - 5 = 0$

Step 1.2.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1

Group the first two terms and the last two terms.

$(6 x^{2} + 3 x) - 10 x - 5 = 0$

Step 1.2.2.2.2

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

$3 x (2 x + 1) - 5 (2 x + 1) = 0$

Step 1.2.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (3 x - 5) = 0$

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 $0$ .

$2 x + 1 = 0$

$3 x - 5 = 0$

Step 1.2.4

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 1.2.4.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 1.2.4.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 1.2.4.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 1.2.5

Set $3 x - 5$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $3 x - 5$ equal to $0$ .

$3 x - 5 = 0$

Step 1.2.5.2

Solve $3 x - 5 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $5$ to both sides of the equation.

$3 x = 5$

Step 1.2.5.2.2

Divide each term in $3 x = 5$ by $3$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $3 x = 5$ by $3$ .

$\frac{3 x}{3} = \frac{5}{3}$

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 $3$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5}{3}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{3}$

Step 1.2.6

The final solution is all the values that make $(2 x + 1) (3 x - 5) = 0$ true.

$x = - \frac{1}{2}, \frac{5}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{1}{2}, 0), (\frac{5}{3}, 0)$

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 $y$ .

$y = 6 {(0)}^{2} - 7 \cdot 0 - 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 6 \cdot 0^{2} - 7 \cdot 0 - 5$

Step 2.2.2

Remove parentheses.

$y = 6 {(0)}^{2} - 7 \cdot 0 - 5$

Step 2.2.3

Simplify $6 {(0)}^{2} - 7 \cdot 0 - 5$ .

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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 $0$ .

$y = 6 \cdot 0 - 7 \cdot 0 - 5$

Step 2.2.3.1.2

Multiply $6$ by $0$ .

$y = 0 - 7 \cdot 0 - 5$

Step 2.2.3.1.3

Multiply $- 7$ by $0$ .

$y = 0 + 0 - 5$

Step 2.2.3.2

Simplify by adding and subtracting.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 - 5$

Step 2.2.3.2.2

Subtract $5$ from $0$ .

$y = - 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 5)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{1}{2}, 0), (\frac{5}{3}, 0)$

y-intercept(s): $(0, - 5)$

Step 4