1 Answer

Djabraham · Answer 1 · 2022-12-16T05:44:22+0000

Answer:

$\displaystyle x_1=(5+√(13))/(2)\approx4.3\text{ and } x_2=(5+√(13))/(2)\approx0.7$

Explanation:

We are given:

x^2-5x+3=0

And we want to solve this via completing the square.

First, we will subtract 3 from both sides. This yields:

x^2-5x=-3

Next, we will factor out the leading coefficient from the left.

In this case, the leading coefficient is simply 1 so we can leave it as is.

Next, we want to add a constant that is half of the b coefficient squared.

In this case, b = -5.

Half of that is -5/2.

Squaring yields 25/4.

So, we will add 25/4 to both sides. Since our leading coefficient is 1, we don't need to multiply by anything when we add it on the right. Hence:

$\displaystyle x^2-5x+(25)/(4)=-3+(25)/(4)$

The left side constitutes a perfect square trinomial as shown:

$\displaystyle (x)^2-2\Big((5)/(2) \Big)(x)+\Big((5)/(2)\Big)^2=-(12)/(4)+(25)/(4)$

Factor and subtract:

$\displaystyle \Big(x-(5)/(2)\Big)^2=(13)/(4)$

Taking the square root of both sides gives:

$\displaystyle x-(5)/(2)=\pm(√(13))/(2)$

Hence:

$\displaystyle x=(5)/(2)\pm(√(13))/(2)}=(5\pm√(13))/(2)$

Approximate:

$\displaystyle x_1=(5+√(13))/(2)\approx4.3\text{ and } x_2=(5-√(13))/(2)\approx0.7$

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