Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.1.2

Rewrite the expression.

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

Step 1.2.3.2.1.2

Multiply $2$ by $- 5$ .

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

Step 1.2.3.2.1.3

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.3.1

Cancel the common factor.

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

Step 1.2.3.2.1.3.2

Rewrite the expression.

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

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $0$ by $2$ .

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

Step 1.2.4

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

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Step 1.2.4.1

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 $- 10$ .

$- 7, - 3$

Step 1.2.4.2

Write the factored form using these integers.

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

Step 1.2.5

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

$x - 7 = 0$

$x - 3 = 0$

Step 1.2.6

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

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Step 1.2.6.1

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

$x - 7 = 0$

Step 1.2.6.2

Add $7$ to both sides of the equation.

$x = 7$

Step 1.2.7

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

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Step 1.2.7.1

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

$x - 3 = 0$

Step 1.2.7.2

Add $3$ to both sides of the equation.

$x = 3$

Step 1.2.8

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

$x = 7, 3$

Step 1.3

x-intercept(s) in point form.

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

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

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

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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 $0$ .

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

Step 2.2.3.1.2

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

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

Step 2.2.3.1.3

Multiply $- 5$ by $0$ .

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

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

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

Step 2.2.3.2.2

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

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

Step 2.3

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