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zmey [24]
2 years ago
15

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

Mathematics
1 answer:
aleksley [76]2 years ago
3 0

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, <em>a</em>, <em>b</em>, and <em>c</em> 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 <em>a</em> 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 <em>x</em> 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 <em>a</em>, <em>b</em>, and <em>c</em> are integers, you can go straight to the answer by simply plugging your values for <em>a</em>, <em>b,</em> and <em>c</em> into the quadratic formulax = -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.

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