Finding a question related to 'finding the square' difficult

I'm currently partaking a distance learning course in A-level mathematics, and have reached chapter 4 in Pure Core Maths 1&2. Found everything pretty straight-forward so far, yet this question I would greatly appreciate help with.

Express 3x² - 12x + 5 in the form A(x - B)² - C :

From the first part of the question I have : 3(x - 2)² -7

I'm then asked to find the minimum value of each of the following expressions:

1. 3x² - 12x + 5 = -7 (The value of x obviously being 2).

And 2. (3x² - 12x +5)²

It's part 2 that's confusing me? Really unsure of where to start.

Re: Finding a question related to 'finding the square' difficult

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Originally Posted by

**DonGorgon** I'm currently partaking a distance learning course in A-level mathematics, and have reached chapter 4 in Pure Core Maths 1&2. Found everything pretty straight-forward so far, yet this question I would greatly appreciate help with.

Express 3x² - 12x + 5 in the form A(x - B)² - C :

From the first part of the question I have : 3(x - 2)² -7

I'm then asked to find the minimum value of each of the following expressions:

1. 3x² - 12x + 5 = -7 (The value of x obviously being 2).

And 2. (3x² - 12x +5)²

It's part 2 that's confusing me? Really unsure of where to start.

Are you allowed to use Calculus?

Re: Finding a question related to 'finding the square' difficult

I haven't covered any calculus yet nope.

Re: Finding a question related to 'finding the square' difficult

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Originally Posted by

**DonGorgon** I haven't covered any calculus yet nope.

Hm, well first notice that $\displaystyle \displaystyle \begin{align*} \left(3x^2 - 12x + 5\right)^2 \end{align*}$, being a square, can never be negative. So it makes sense that the minimum value would be 0. Let's try to evaluate where the function will be 0...

$\displaystyle \displaystyle \begin{align*} \left(3x^2 - 12x + 5\right)^2 &= 0 \\ 3x^2 - 12x + 5 &= 0 \\ x &= \frac{12 \pm \sqrt{(-12)^2 - 4(3)(5)}}{2(3)} \\ x &= \frac{12 \pm \sqrt{144 - 60}}{6} \\ x &= \frac{12 \pm \sqrt{84}}{6} \\ x &= \frac{12 \pm 2\sqrt{21}}{6} \\ x &= \frac{6 \pm \sqrt{21}}{3}\end{align*}$

So the minimum value is 0, which occurs when $\displaystyle \displaystyle \begin{align*} x = \frac{6 - \sqrt{21}}{3} \end{align*}$ or $\displaystyle \displaystyle \begin{align*} \frac{6 + \sqrt{21}}{3} \end{align*}$

Re: Finding a question related to 'finding the square' difficult

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**Prove It** Hm, well first notice that $\displaystyle \displaystyle \begin{align*} \left(3x^2 - 12x + 5\right)^2 \end{align*}$, being a square, can never be negative. So it makes sense that the minimum value would be 0. Let's try to evaluate where the function will be 0...

$\displaystyle \displaystyle \begin{align*} \left(3x^2 - 12x + 5\right)^2 &= 0 \\ 3x^2 - 12x + 5 &= 0 \\ x &= \frac{12 \pm \sqrt{(-12)^2 - 4(3)(5)}}{2(3)} \\ x &= \frac{12 \pm \sqrt{144 - 60}}{6} \\ x &= \frac{12 \pm \sqrt{84}}{6} \\ x &= \frac{12 \pm 2\sqrt{21}}{6} \\ x &= \frac{6 \pm \sqrt{21}}{3}\end{align*}$

So the minimum value is 0, which occurs when $\displaystyle \displaystyle \begin{align*} x = \frac{6 - \sqrt{21}}{3} \end{align*}$ or $\displaystyle \displaystyle \begin{align*} \frac{6 + \sqrt{21}}{3} \end{align*}$

Thanks for getting back to me. Ah ok, from my attempt I used the expression in the form A(x-B)² - C :

3(x - 2)² - 7 = 0

3(x - 2)² = 7

3(x - 2) = ± √7

(x - 2) = ± √7 / 3

x = 2 ± √7 / 3

Re: Finding a question related to 'finding the square' difficult

Quote:

Originally Posted by

**DonGorgon** Thanks for getting back to me. Ah ok, from my attempt I used the expression in the form A(x-B)² - C :

3(x - 2)² - 7 = 0

3(x - 2)² = 7

3(x - 2) = ± √7

(x - 2) = ± √7 / 3

x = 2 ± √7 / 3

By the order of operations, exponentiation is done before multiplication. So when you go in reverse and solve for x, you need to undo the multiplication first.