Show proof of deriving the Quadratic Formula from the Quadratic Equation ax^2 + bx + c = )
To derive the quadratic formula from the quadratic equation ax^2 + bx + c = 0, we can follow these steps:
Step 1: Start with the quadratic equation: ax^2 + bx + c = 0.
Step 2: Divide the entire equation by a (the coefficient of x^2) to simplify it:
x^2 + (b/a)x + (c/a) = 0.
Step 3: Move the constant term to the other side of the equation:
x^2 + (b/a)x = - (c/a).
Step 4: Complete the square on the left side of the equation. To do this, take half of the coefficient of x, square it, and add it to both sides of the equation. In this case, the coefficient of x is (b/a), so half of it is (b/2a).
x^2 + (b/a)x + (b/2a)^2 = - (c/a) + (b/2a)^2.
Simplifying this, we get:
x^2 + (b/a)x + (b^2)/(4a^2) = - (c/a) + (b^2)/(4a^2).
Step 5: On the left side, factor the perfect square trinomial:
(x + (b/2a))^2 = - (c/a) + (b^2)/(4a^2).
Step 6: Simplify the equation on the right side:
(x + (b/2a))^2 = (b^2 - 4ac)/(4a^2).
Step 7: Take the square root of both sides of the equation to isolate x:
x + (b/2a) = ± √((b^2 - 4ac)/(4a^2)).
Step 8: Simplify the equation on the right side:
x + (b/2a) = ± (√(b^2 - 4ac))/(2a).
Step 9: Finally, isolate x by subtracting (b/2a) from both sides of the equation:
x = (-b ± √(b^2 - 4ac))/(2a).
This is the quadratic formula derived from the quadratic equation ax^2 + bx + c = 0.
To derive the quadratic formula from the quadratic equation ax^2 + bx + c = 0, we can use the method of completing the square, which involves manipulating the equation algebraically.
Step 1: Start with the given quadratic equation:
ax^2 + bx + c = 0
Step 2: Divide the entire equation by 'a' to make the coefficient of x^2 equal to 1 (assuming 'a' is not zero):
x^2 + (b/a)x + c/a = 0
Step 3: Move the constant term (c/a) to the right side of the equation:
x^2 + (b/a)x = -c/a
Step 4: Take the coefficient of x (b/a) and divide it by 2, then square the result [(b/2a)^2]:
(b/2a)^2 = b^2/(4a^2)
Step 5: Add and subtract the result obtained in Step 4 from the left side of the equation:
x^2 + (b/a)x + b^2/(4a^2) - b^2/(4a^2) = -c/a
Step 6: Rewrite the first three terms as a perfect square:
(x + b/(2a))^2 = b^2/(4a^2) - c/a
Step 7: Simplify the right side of the equation:
(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)
Step 8: Take the square root of both sides of the equation:
x + b/(2a) = ±√[(b^2 - 4ac)/(4a^2)]
Step 9: Isolate the variable x by subtracting b/(2a) from both sides of the equation:
x = (-b ± √(b^2 - 4ac))/(2a)
This is the quadratic formula, which gives the solutions to the quadratic equation ax^2 + bx + c = 0. The ± symbol indicates that there are two possible solutions, one with a positive sign and the other with a negative sign.