Question

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

Answer 1

Answer: $x = -b\frac{+}{-} \sqrt{b^{2} } - 4ac$ 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} +\frac{b}{a}x + \frac{c}{a} = 0$$.$

Next we move the $\frac{c}{a}$ to the right side of the equation by subtracting $\frac{c}{a}$ from both sides and we have $x^{2} + \frac{b}{a}x = \frac{-c}{a}.$ 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, $+\frac{b}{a},$ we can take half of it by simply doubling the denominator to get $+\frac{b}{2a}$ and when squaring $+\frac{b}{2a},$ remember to square both the numerator and denominator so we get $+\frac{b^{2} }{4a^{2}}.$

So we add $\frac{b^{2} }{4a^{2} }$ 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 $+\frac{b}{a}$ which is $+\frac{b}{2a}$ and we have $(x +\frac{b}{2^{a}})^{2}$.

On the right, when adding $-\frac{c}{a} + \frac{b^{2} }{4a^{2} }$, our common denominator is $4a^{2}$ so we multiply top and bottom of $-\frac{c}{a}$ by $4a$ to get $\frac{-4ac}{4a^{2} } + \frac{b^{2} }{4a^{2} }$ which we can rewrite as $\frac{b^{2}- 4ac}{4a^{2} }.$ 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 $x + \frac{b}{2a}$ 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 $\frac{+}{-} \sqrt{b^{2} } - 4ac$ over 2a.

To get x by itself, subtract $\frac{b}{2a}$ from both sides and we have

$x = -b\frac{+}{-} \sqrt{b^{2} } - 4ac$ 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$x = -b\frac{+}{-} \sqrt{b^{2} } - 4ac$ 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.