Solve the following quadratic equation by completing the square. Simplify the solutions and rationalize denominators, if necessary.
x2 + 6x − 17 = 0
x^2 + 6x - 17 = 0
First separate x terms
X^2 + 6x = 17
or
17 = x^2 + 6x
remember that aX^2 + bx + c
to complete square, divide b by two, and square it.
6/2 = 3.
3^2 = 9.
Now add 9 to each side of the equation
17+9 = x^2 + 6x + 9
combine like terms:
26 = x^2 + 6x + 9
Now factor the right side:
26 = (x+3)(x+3)
26 = (x+3)^2
Now square root the right and left side to get rid of the square:
+,-sqroot(26) = x+3
Your remaining solutions are:
x = -3 +sqr(26)
x = -3 -sqr(26)
note: please recheck into equation, they might not be correct!
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Move the constant term (-17) to the right side of the equation, leaving the x² and x terms on the left:
x² + 6x = 17
Step 2: Take half of the coefficient of the x-term (6) and square it:
(6/2)² = 9
Step 3: Add the square obtained in step 2 to both sides of the equation:
x² + 6x + 9 = 17 + 9
Simplifying gives:
x² + 6x + 9 = 26
Step 4: Rewrite the left side of the equation as the square of a binomial:
(x + 3)² = 26
Step 5: Take the square root of both sides of the equation:
√((x + 3)²) = ±√26
Taking the square root gives:
x + 3 = ±√26
Step 6: Solve for x:
For x + 3 = √26, subtract 3 from both sides:
x = -3 + √26
For x + 3 = -√26, subtract 3 from both sides:
x = -3 - √26
So the solutions to the given quadratic equation are:
x = -3 + √26
x = -3 - √26
To rationalize the denominators, multiply both the numerator and denominator of the second solution by √26:
x = (-3 - √26)(√26/√26)
x = (-3√26 - 26)/26
Hence, the simplified and rationalized solutions to the quadratic equation x² + 6x - 17 = 0 are:
x = -3 + √26
x = (-3√26 - 26)/26