Convert to Interval Notation x^2-10x+25<0

$x^{2} - 10 x + 25 < 0$

Step 1

Convert the inequality to an equation.

$x^{2} - 10 x + 25 = 0$

Step 2

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

$x^{2} - 10 x + 5^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

${(x - 5)}^{2} = 0$

Step 3

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4

Add $5$ to both sides of the equation.

$x = 5$

Step 5

Use each root to create test intervals.

$x < 5$

$x > 5$

Step 6

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 $x < 5$ to see if it makes the inequality true.

Choose a value on the interval $x < 5$ and see if this value makes the original inequality true.

$x = 0$

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

${(0)}^{2} - 10 \cdot 0 + 25 < 0$

The left side $25$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 5$ to see if it makes the inequality true.

Choose a value on the interval $x > 5$ and see if this value makes the original inequality true.

$x = 8$

Replace $x$ with $8$ in the original inequality.

${(8)}^{2} - 10 \cdot 8 + 25 < 0$

The left side $9$ is not less than the right side $0$ , which means that the given statement is false.

False

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

$x < 5$ False

$x > 5$ False

$x < 5$ False

$x > 5$ False

Step 7

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Convert to Interval Notation x^2-10x+25<0

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