Find the x and y Intercepts f(x)=9x^2-5x+2

$f (x) = 9 x^{2} - 5 x + 2$

Step 1

Find the x-intercepts.

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

$0 = 9 x^{2} - 5 x + 2$

Solve the equation.

Rewrite the equation as $9 x^{2} - 5 x + 2 = 0$ .

$9 x^{2} - 5 x + 2 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 9$ , $b = - 5$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (9 \cdot 2)}}{2 \cdot 9}$

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 2$ .

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36 \cdot 2}}{2 \cdot 9}$

Multiply $- 36$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 72}}{2 \cdot 9}$

Subtract $72$ from $25$ .

$x = \frac{5 \pm \sqrt{- 47}}{2 \cdot 9}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 9}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{47}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{5 \pm i \sqrt{47}}{18}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 2$ .

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36 \cdot 2}}{2 \cdot 9}$

Multiply $- 36$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 72}}{2 \cdot 9}$

Subtract $72$ from $25$ .

$x = \frac{5 \pm \sqrt{- 47}}{2 \cdot 9}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 9}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{47}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{5 \pm i \sqrt{47}}{18}$

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{47}}{18}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}$

Multiply $- 4 \cdot 9 \cdot 2$ .

Multiply $- 4$ by $9$ .

$x = \frac{5 \pm \sqrt{25 - 36 \cdot 2}}{2 \cdot 9}$

Multiply $- 36$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 72}}{2 \cdot 9}$

Subtract $72$ from $25$ .

$x = \frac{5 \pm \sqrt{- 47}}{2 \cdot 9}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 47}}{2 \cdot 9}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot 9}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{47}}{2 \cdot 9}$

Multiply $2$ by $9$ .

$x = \frac{5 \pm i \sqrt{47}}{18}$

Change the $\pm$ to $-$ .

$x = \frac{5 - i \sqrt{47}}{18}$

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{47}}{18}, \frac{5 - i \sqrt{47}}{18}$

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

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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

$y = 9 {(0)}^{2} - 5 \cdot 0 + 2$

Solve the equation.

Remove parentheses.

$y = 9 \cdot 0^{2} - 5 \cdot 0 + 2$

Remove parentheses.

$y = 9 {(0)}^{2} - 5 \cdot 0 + 2$

Simplify $9 {(0)}^{2} - 5 \cdot 0 + 2$ .

Simplify each term.

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

$y = 9 \cdot 0 - 5 \cdot 0 + 2$

Multiply $9$ by $0$ .

$y = 0 - 5 \cdot 0 + 2$

Multiply $- 5$ by $0$ .

$y = 0 + 0 + 2$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 2$

Add $0$ and $2$ .

$y = 2$

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 2)$

Step 4

Find the x and y Intercepts f(x)=9x^2-5x+2

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