Find the Domain y = square root of x^2-3x+2

$y = \sqrt{x^{2} - 3 x + 2}$

Step 1

Set the radicand in $\sqrt{x^{2} - 3 x + 2}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} - 3 x + 2 \geq 0$

Step 2

Solve for $x$ .

Convert the inequality to an equation.

$x^{2} - 3 x + 2 = 0$

Factor $x^{2} - 3 x + 2$ 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 $2$ and whose sum is $- 3$ .

$- 2, - 1$

Write the factored form using these integers.

$(x - 2) (x - 1) = 0$

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

$x - 2 = 0$

$x - 1 = 0$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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

$x = 2, 1$

Use each root to create test intervals.

$x < 1$

$1 < x < 2$

$x > 2$

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < 1$ to see if it makes the inequality true.

Choose a value on the interval $x < 1$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

${(0)}^{2} - 3 \cdot 0 + 2 \geq 0$

The left side $2$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $1 < x < 2$ and see if this value makes the original inequality true.

$x = 1.5$

Replace $x$ with $1.5$ in the original inequality.

${(1.5)}^{2} - 3 \cdot 1.5 + 2 \geq 0$

The left side $- 0.25$ is less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

${(4)}^{2} - 3 \cdot 4 + 2 \geq 0$

The left side $6$ is greater than the right side $0$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1$ True

$1 < x < 2$ False

$x > 2$ True

$x < 1$ True

$1 < x < 2$ False

$x > 2$ True

The solution consists of all of the true intervals.

$x \leq 1$ or $x \geq 2$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 1] \cup [2, \infty)$

Set-Builder Notation:

${x | x \leq 1, x \geq 2}$

Step 4

Find the Domain y = square root of x^2-3x+2

Related Questions

Support LCMGCF.com by disabling your adblocker.