Find the x and y Intercepts f(x)=1/2x^2-5x+21/2

$f (x) = \frac{1}{2} x^{2} - 5 x + \frac{21}{2}$

Step 1

Find the x-intercepts.

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

$0 = \frac{1}{2} x^{2} - 5 x + \frac{21}{2}$

Solve the equation.

Rewrite the equation as $\frac{1}{2} x^{2} - 5 x + \frac{21}{2} = 0$ .

$\frac{1}{2} x^{2} - 5 x + \frac{21}{2} = 0$

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} - 5 x + \frac{21}{2} = 0$

Multiply each term in $\frac{x^{2}}{2} - 5 x + \frac{21}{2} = 0$ by $2$ to eliminate the fractions.

Multiply each term in $\frac{x^{2}}{2} - 5 x + \frac{21}{2} = 0$ by $2$ .

$\frac{x^{2}}{2} \cdot 2 - 5 x \cdot 2 + \frac{21}{2} \cdot 2 = 0 \cdot 2$

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x^{2}}{2} \cdot 2 - 5 x \cdot 2 + \frac{21}{2} \cdot 2 = 0 \cdot 2$

Rewrite the expression.

$x^{2} - 5 x \cdot 2 + \frac{21}{2} \cdot 2 = 0 \cdot 2$

Multiply $2$ by $- 5$ .

$x^{2} - 10 x + \frac{21}{2} \cdot 2 = 0 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} - 10 x + \frac{21}{2} \cdot 2 = 0 \cdot 2$

Rewrite the expression.

$x^{2} - 10 x + 21 = 0 \cdot 2$

Simplify the right side.

Multiply $0$ by $2$ .

$x^{2} - 10 x + 21 = 0$

Factor $x^{2} - 10 x + 21$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $21$ and whose sum is $- 10$ .

$- 7, - 3$

Write the factored form using these integers.

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

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

$x - 7 = 0$

$x - 3 = 0$

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

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

$x = 7, 3$

x-intercept(s) in point form.

x-intercept(s): $(7, 0), (3, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{1}{2} \cdot {(0)}^{2} - 5 \cdot 0 + \frac{21}{2}$

Solve the equation.

Remove parentheses.

$y = \frac{1}{2} \cdot 0^{2} - 5 \cdot 0 + \frac{21}{2}$

Remove parentheses.

$y = \frac{1}{2} \cdot {(0)}^{2} - 5 \cdot 0 + \frac{21}{2}$

Simplify $\frac{1}{2} \cdot {(0)}^{2} - 5 \cdot 0 + \frac{21}{2}$ .

Simplify each term.

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

$y = \frac{1}{2} \cdot 0 - 5 \cdot 0 + \frac{21}{2}$

Multiply $\frac{1}{2}$ by $0$ .

$y = 0 - 5 \cdot 0 + \frac{21}{2}$

Multiply $- 5$ by $0$ .

$y = 0 + 0 + \frac{21}{2}$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + \frac{21}{2}$

Add $0$ and $\frac{21}{2}$ .

$y = \frac{21}{2}$

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{21}{2})$

Step 3

List the intersections.

x-intercept(s): $(7, 0), (3, 0)$

y-intercept(s): $(0, \frac{21}{2})$

Step 4

Find the x and y Intercepts f(x)=1/2x^2-5x+21/2

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