Answer: [tex]x = -b\frac{+}{-} \sqrt{b^{2} } - 4ac[/tex] 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 [tex]x^{2} +\frac{b}{a}x + \frac{c}{a} = 0[/tex][tex].[/tex]
Next we move the [tex]\frac{c}{a}[/tex] to the right side of the equation by subtracting [tex]\frac{c}{a}[/tex] from both sides and we have [tex]x^{2} + \frac{b}{a}x = \frac{-c}{a}.[/tex] 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, [tex]+\frac{b}{a},[/tex] we can take half of it by simply doubling the denominator to get [tex]+\frac{b}{2a}[/tex] and when squaring [tex]+\frac{b}{2a},[/tex] remember to square both the numerator and denominator so we get [tex]+\frac{b^{2} }{4a^{2}}.[/tex]
So we add [tex]\frac{b^{2} }{4a^{2} }[/tex] 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 [tex]+\frac{b}{a}[/tex] which is [tex]+\frac{b}{2a}[/tex] and we have [tex](x +\frac{b}{2^{a}})^{2}[/tex].
On the right, when adding [tex]-\frac{c}{a} + \frac{b^{2} }{4a^{2} }[/tex], our common denominator is [tex]4a^{2}[/tex] so we multiply top and bottom of [tex]-\frac{c}{a}[/tex] by [tex]4a[/tex] to get [tex]\frac{-4ac}{4a^{2} } + \frac{b^{2} }{4a^{2} }[/tex] which we can rewrite as [tex]\frac{b^{2}- 4ac}{4a^{2} }.[/tex] 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 [tex]x + \frac{b}{2a}[/tex] 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 [tex]\frac{+}{-} \sqrt{b^{2} } - 4ac[/tex] over 2a.
To get x by itself, subtract [tex]\frac{b}{2a}[/tex] from both sides and we have
[tex]x = -b\frac{+}{-} \sqrt{b^{2} } - 4ac[/tex] 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[tex]x = -b\frac{+}{-} \sqrt{b^{2} } - 4ac[/tex] 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.
