How do you derive the quadratic formula? Thanks

Answer 1

See below.

The #color(blue)"Quadratic Formula"# can derived by completing the square algebraically.

From the form:

#ax^2+bx+c#

Start with:

#ax^2+bx+c=0#
Move the constant #c# to the left-hand side:
#ax^2+bx=-c#
Divide by the coefficient of #x^2#:
#a/ax^2+b/ax=-c/a#
#x^2+b/ax=-c/a#
Add to both sides the square of half the coefficient of #x#:
#x^2+b/ax+(b/(2a))^2=-c/a+(b/(2a))^2#

Simplify left hand side:

#x^2+b/ax+b/(2a)=-c/a+b^2/(4a^2)#
#x^2+b/ax+b/(2a)=(b^2-4ac)/(4a^2)#

Arrange right hand side into the square of a binomial:

#(x^2+b/(2a))^2=(b^2-4ac)/(4a^2)#

Take square roots of both sides:

#x+b/(2a)=sqrt((b^2-4ac))/sqrt((4a^2))=+-sqrt((b^2-4ac))/(2a)#
Subtract #b/(2a)# from both sides:
#x=-b/(2a)+-sqrt((b^2-4ac))/(2a)#

Add fractions:

#color(blue)(x=(-b+-sqrt((b^2-4ac)))/(2a))#
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Answer 2

The quadratic formula is derived using the method of completing the square. Start with a general quadratic equation in the form ax^2 + bx + c = 0. Divide both sides by a to simplify the equation to x^2 + (b/a)x + c/a = 0. Move the constant term to the other side to isolate the terms involving x. Complete the square on the x terms by adding and subtracting (b/2a)^2 inside the parentheses. This creates a perfect square trinomial. Factor the perfect square trinomial and then solve for x to get the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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