Simplifying a quadratic equation

[rochey]

I need to simplify the following quadratic equation to allow me to plot the parabola on a graph, however I do not think i have correctly multiplied out the fractions.

y = -x²/5 + x/5 + 6

I have simplified to 5y =- x² + x + 30 but this does not look correct to me if I want to plot a parabola

 

[Deleted member 4993]

I need to simplify the following quadratic equation to allow me to plot the parabola on a graph, however I do not think i have correctly multiplied out the fractions.

y = -x²/5 + x/5 + 6

I have simplified to 5y =- x² + x + 30 but this does not look correct to me if I want to plot a parabola

You have performed the operation of multiplication correctly - and it'll provide a plot of parabola correctly (given that you calculated the co-ordinates correctly) .

My concern is - why do you want to avoid "fractions" (some hockey-players do that - they have been fractured too often). But in your case - I do not see any advantage - as you'll be using a calculator to find the co-ordinates.

 

[HallsofIvy]

Again, yes, you have multiplied both sides by 5 correctly, but why do you want to do that? If your purpose is graph the parabola, it would make more sense just to factor -1/5 out of the first two terms on the right: \(\displaystyle y=-\frac{1}{5}(x^2- x)+ 6\). Now complete the square inside the parentheses.
he coefficient of x is -1, half of that is -1/2 and \(\displaystyle (-1/2)^2=1/4\) so we need to add and subtract \(\displaystyle \frac{1}{4}\):
\(\displaystyle y= -\frac{1}{5}(x^2+ x)+ 6= -\frac{1}{5}(x^2- x+ \frac{1}{4}- \frac{1}{4})+ 6= -\frac{1}{5}((x- \frac{1}{2})^2- \frac{1}{4})+ 6= -\frac{1}{5}(x-\frac{1}{2})^2 + \frac{1}{20}+ 6\)

\(\displaystyle \displaystyle{y=-\frac{1}{5}\left(x-\frac{1}{2}\right)^2+ \frac{121}{20}}\)


That tells you immediately where the vertex of the parabola is.

 

[rochey]

I need to find the x and y intercepts, my thinking behind simplifying was to make it easier to find these by multiplying out the fractions

 

[pappus]

Again, yes, you have multiplied both sides by 5 correctly, but why do you want to do that? If your purpose is graph the parabola, it would make more sense just to factor -1/5 out of the first two terms on the right: \(\displaystyle y=-\frac{1}{5}(x^2- x)+ 6\). Now complete the square inside the parentheses.
he coefficient of x is -1, half of that is -1/2 and \(\displaystyle (-1/2)^2=1/4\) so we need to add and subtract \(\displaystyle \frac{1}{4}\):
\(\displaystyle y= -\frac{1}{5}(x^2+ x)+ 6= -\frac{1}{5}(x^2- x+ \frac{1}{4}- \frac{1}{4})+ 6= -\frac{1}{5}((x- \frac{1}{2})^2- \frac{1}{4})+ 6= -\frac{1}{5}(x-\frac{1}{2})^2- \frac{5}{4}+ 6\)
That tells you immediately where the vertex of the parabola is.

@HallsofIvy: Could it be that there is a small mistake in your calculations? In my opinion the last step should read:

\(\displaystyle \displaystyle{-\frac{1}{5}(x-\frac{1}{2})^2+ \frac{1}{20}+ 6} \)

So you finally get:

\(\displaystyle \displaystyle{y=-\frac{1}{5}\left(x-\frac{1}{2}\right)^2+ \frac{121}{20}} \)

 

[Deleted member 4993]

I need to find the x and y intercepts, my thinking behind simplifying was to make it easier to find these by multiplying out the fractions

Perhaps....

But first apply what is the definition of y-intercept (y_i) → where x = 0 → y_i = -0²/5 + 0/5 + 6 = 6

Then for x-intercept (x_i)y = 0 → 0 = -x_i² /5 + x_i/5 + 6 → Now get rid of the fraction and find x-intercepts →

x_i² - x_i - 30 = 0 →

(x_i - 6) * (x_i + 5) = 0 .... and continue

 

[mmm4444bot]

I need to simplify the following quadratic equation to allow me to plot the parabola

The y-intercept is obvious by inspection; no paper and pencil are required for that.

The Quadratic Formula is another method to determine the x-intercepts. No "simplification" of the given equation is necessary for that.

Once the intercepts are known, calculating the vertex is easy.

OPINION: I think that all of this will take no more than 10 minutes, for people who (1) know what the Quadratic Formula is and how to apply it, (2) have previously memorized the multiplication table through 12×12, and (3) have the ability to complete simple arithmetic with small fractions. :cool:

 

[HallsofIvy]

@HallsofIvy: Could it be that there is a small mistake in your calculations? In my opinion the last step should read:

\(\displaystyle \displaystyle{-\frac{1}{5}(x-\frac{1}{2})^2+ \frac{1}{20}+ 6} \)

So you finally get:

\(\displaystyle \displaystyle{y=-\frac{1}{5}\left(x-\frac{1}{2}\right)^2+ \frac{121}{20}} \)

It's my version of the new math: \(\displaystyle \frac{1}{5}\frac{1}{4}= \frac{4}{5}\)! You don't think it will catch on?

(Thanks for the correction.)

 

[pappus]

It's my version of the new math: \(\displaystyle \frac{1}{5}\frac{1}{4}= \frac{4}{5}\)! You don't think it will catch on?

(Thanks for the correction.)

Your New Math can be installed quite easily, you only have to assume that \(\displaystyle \frac00 = 1 \)

But by first inspection your New Math is already used by European bankers and politicians.

 

[mmm4444bot]

\(\displaystyle \frac{1}{5}\frac{1}{4}= \frac{4}{5}\)

But you posted 5/4.

Must still be too new.