Solve the quadratic equation ax2+bx+c=0 using the completing the squares
To solve the quadratic equation ax^2 + bx + c = 0 using the method of completing the square, follow these steps:
Step 1: Make sure the coefficient 'a' is equal to 1 by dividing the entire equation by 'a'.
Divide every term by 'a': x^2 + (b/a)x + c/a = 0
Step 2: Move the constant term 'c/a' to the right side of the equation.
Rearrange the equation: x^2 + (b/a)x = -c/a
Step 3: Take half of the coefficient of 'x', square it, and add it to both sides of the equation.
Add (b/2a)^2 on both sides: x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2
Step 4: Simplify the right side of the equation.
Simplify: x^2 + (b/a)x + (b/2a)^2 = (-4ac + b^2) / (4a^2)
Step 5: Factor the left side of the equation.
Write the left side as a perfect square: (x + b/2a)^2 = (-4ac + b^2) / (4a^2)
Step 6: Take the square root of both sides of the equation.
Take the square root: x + b/2a = ±√((-4ac + b^2) / (4a^2))
Step 7: Isolate 'x' on one side of the equation.
Solve for 'x': x = -(b/2a) ± √((-4ac + b^2) / (4a^2))
The solutions to the quadratic equation ax^2 + bx + c = 0 can be obtained by plugging in the values of 'a', 'b', and 'c' into the equation x = -(b/2a) ± √((-4ac + b^2) / (4a^2)).
ax2+bx+c=0
ax^2 + bx = -c <--- subtracted c from both sides
x^2 + (b/a)x = -c/a <--- divide everybody by a
x^2 + (b/a)x + b^2/(4a^2) = -c/a + b^2/(4a^2) <--- "completed the square"
(x + b/(2a) )^2 = (b^2 - 4ac)/(4a^2) <---simplified
x + b/(2a) = ±√(b^2 - 4ac)/(2a) <--took √ of both sides
x = -b/(2a) ± √(b^2-4ac)/(2a)
x = ( -b ± √(b^2 - 4ac)/(2a) <-- solved for x and simplified
Beyold the infamous quadratic equation formula