Find the x and y Intercepts p(x)=2x^2-x-21

$p (x) = 2 x^{2} - x - 21$

Step 1

Find the x-intercepts.

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

$0 = 2 x^{2} - x - 21$

Solve the equation.

Rewrite the equation as $2 x^{2} - x - 21 = 0$ .

$2 x^{2} - x - 21 = 0$

Factor by grouping.

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 $a \cdot c = 2 \cdot - 21 = - 42$ and whose sum is $b = - 1$ .

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

$2 x^{2} - (x) - 21 = 0$

Rewrite $- 1$ as $6$ plus $- 7$

$2 x^{2} + (6 - 7) x - 21 = 0$

Apply the distributive property.

$2 x^{2} + 6 x - 7 x - 21 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} + 6 x) - 7 x - 21 = 0$

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

$2 x (x + 3) - 7 (x + 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

$(x + 3) (2 x - 7) = 0$

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

$x + 3 = 0$

$2 x - 7 = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

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

$2 x - 7 = 0$

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

Add $7$ to both sides of the equation.

$2 x = 7$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

The final solution is all the values that make $(x + 3) (2 x - 7) = 0$ true.

$x = - 3, \frac{7}{2}$

x-intercept(s) in point form.

x-intercept(s): $(- 3, 0), (\frac{7}{2}, 0)$

Step 2

Find the y-intercepts.

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

$y = 2 {(0)}^{2} - (0) - 21$

Solve the equation.

Remove parentheses.

$y = 2 \cdot 0^{2} - (0) - 21$

Remove parentheses.

$y = 2 {(0)}^{2} - (0) - 21$

Simplify $2 {(0)}^{2} - (0) - 21$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = 2 \cdot 0 - (0) - 21$

Multiply $2$ by $0$ .

$y = 0 - (0) - 21$

Multiply $- 1$ by $0$ .

$y = 0 + 0 - 21$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 21$

Subtract $21$ from $0$ .

$y = - 21$

y-intercept(s) in point form.

y-intercept(s): $(0, - 21)$

Step 3

List the intersections.

x-intercept(s): $(- 3, 0), (\frac{7}{2}, 0)$

y-intercept(s): $(0, - 21)$

Step 4

Find the x and y Intercepts p(x)=2x^2-x-21

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