Question

solve the quadratic equation by completing the square.
$x^2 - 10x + 19 = 0$
first, choose the appropriate form and fill in the blanks with the correct numbers.
then, solve the equation. round your answer to the nearest hundredth.
if there is more than one solution, separate them with commas.
form:
$circ (x + square)^2 = square$
$circ (x - square)^2 = square$
solution:
$x = square$

Explanation:

Step1: Move constant to right

Given \(x^2 - 10x + 19 = 0\), subtract 19: \(x^2 - 10x = -19\).

Step2: Complete the square

Take half of -10, square it: \((\frac{-10}{2})^2 = 25\). Add to both sides: \(x^2 - 10x + 25 = -19 + 25\).

Step3: Factor left side

\((x - 5)^2 = 6\).

Step4: Solve for x

Take square roots: \(x - 5 = \pm\sqrt{6}\). Then \(x = 5 \pm\sqrt{6}\).

Step5: Calculate decimal values

\(\sqrt{6} \approx 2.45\), so \(x = 5 + 2.45 = 7.45\) or \(x = 5 - 2.45 = 2.55\).

Answer:

Form: \((x - 5)^2 = 6\)
Solution: \(x = 7.45, 2.55\)