Question

Solve for x, ax^2+bx+c=0

Answer 1

Answer:
`x = -b(+)/(-) \sqrt{b^(2) } - 4ac` over 2a

Step-by-step explanation: To solve this problem, we will use the method of completing the square.

In this problem, a, b, and c are integers. So to complete the square, it's important to understand that we can't have a coefficient on the x² term so we divide both sides of the equation by a to get
`x^(2) +(b)/(a)x + (c)/(a) = 0`
`.`

Next we move the
`(c)/(a)` to the right side of the equation by subtracting
`(c)/(a)` from both sides and we have
`x^(2) + (b)/(a)x = (-c)/(a).` To complete the square, it's important to understand that we take half the coefficient of the middle terms squared. Since the coefficient of the middle term is a fraction,
`+(b)/(a),` we can take half of it by simply doubling the denominator to get
`+(b)/(2a)` and when squaring
`+(b)/(2a),` remember to square both the numerator and denominator so we get
`+(b^(2) )/(4a^(2)).`

So we add
`(b^(2) )/(4a^(2) )` to both sides of the equation. Next, remember that our trinomial on the left factors as a binomial squared and the binomial uses half the coefficient of the middle term of the trinomial. So we use half of
`+(b)/(a)` which is
`+(b)/(2a)` and we have
`(x +(b)/(2^(a)))^(2)`.

On the right, when adding
`-(c)/(a) + (b^(2) )/(4a^(2) )`, our common denominator is
`4a^(2)` so we multiply top and bottom of
`-(c)/(a)` by
`4a` to get
`(-4ac)/(4a^(2) ) + (b^(2) )/(4a^(2) )` which we can rewrite as
`(b^(2)- 4ac)/(4a^(2) ).` Note that we have switched the order of the -4ac and the b². Don't get thrown off here.

To get rid of the square on the left side of the equation, we square root both sides so on the left we have
`x + (b)/(2a)` and on the right remember to use + or - and also remember that when square rooting a fraction, we must square root both the numerator and the denominator so we have
`(+)/(-) \sqrt{b^(2) } - 4ac` over 2a.

To get x by itself, subtract
`(b)/(2a)` from both sides and we have

`x = -b(+)/(-) \sqrt{b^(2) } - 4ac` over 2a.

Remember the answer to this problem. It's called the quadratic formula. The beauty of the quadratic formula is as long as your quadratic is in the form

ax² + bx + c = 0 where a, b, and c are integers, you can go straight to the answer by simply plugging your values for a, b, and c into the quadratic formula
`x = -b(+)/(-) \sqrt{b^(2) } - 4ac` over 2a.

It's great to memorize this formula as it is used in many problems.

I have also attached my work in the image provided.