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If 3x^2-2x+7=0,then (x-1/3)^2= please help with detailed steps because i dont really understand it. I know the answer is -20/9 but please explain

Sagot :

For this case we have the following polynomial:

[tex]3x^2-2x+7=0[/tex]

To solve the problem, we must complete squares.

The first step is to divide the entire expression by 3.

We have then:

[tex]\frac{3}{3}x^2-\frac{2}{3}x+\frac{7}{3}=0[/tex]

The second step is to place the constant term on the right side of the equation:

[tex] \frac{3}{3}x^2-\frac{2}{3}x=-\frac{7}{3} [/tex]

The third step is to complete the square:

[tex] \frac{3}{3}x^2-\frac{2}{3}x + (-\frac{1}{3})^2=-\frac{7}{3}+ (-\frac{1}{3})^2 [/tex]

Rewriting we have:

[tex] x^2-\frac{2}{3}x + \frac{1}{9}=-\frac{7}{3}+ \frac{1}{9} [/tex]

[tex] (x-\frac{1}{3})^2 = -\frac{20}{3} [/tex]

Answer:

By completing squares we have:

[tex] (x-\frac{1}{3})^2 = -\frac{20}{3} [/tex]

Answer:

[tex]-\dfrac{20}{9}[/tex]

Explanation:

A quadratic function is a kind of function with highest degree 2 .  Standard form of the quadratic equation : tex]ax^2+bx+c=0[/tex]

Further explanation:

Consider the given quadratic equation : [tex]3x^2-2x+7=0[/tex]  

First we divide both sides  by 3 , we get

[tex]x^2-\dfrac{2}{3}x+\dfrac{7}{3}=0[/tex]--------(1)

Compare this equation to [tex]x^2+2ax+a^2[/tex] , we have

[tex]2a=\dfrac{-2}{3}[/tex]  

[tex]\Rightarrow\ a=\dfrac{-1}{3}[/tex]  [divide both sides by 2]

Now using the completing the squares method , Add and subtract [tex](\dfrac{-1}{3})^2[/tex] to the left side in (1), we get

[tex]x^2-\dfrac{2}{3}x+(\dfrac{-1}{3})^2-(\dfrac{-1}{3})^2+\dfrac{7}{3}=0[/tex]  

It can be written as

[tex](x^2-2(\dfrac{1}{3})x+\dfrac{1}{3})^2)-\dfrac{1}{9}+\dfrac{7}{3}=0[/tex]  

Use identity [tex]x^2-2ax+a^2=(x-a)^2[/tex], we have

[tex](x-\dfrac{1}{3})^2)+\dfrac{7(3)-1}{9}=0[/tex]  

[tex](x-\dfrac{1}{3})^2)+\dfrac{20}{9}=0[/tex]  

Subtract [tex]\dfrac{20}{9}[/tex] from both the sides , we get

[tex](x-\dfrac{1}{3})^2)=-\dfrac{20}{9}[/tex]

Therefore, the value of [tex](x-\dfrac{1}{3})^2)=-\dfrac{20}{9}[/tex]

Learn more :

  • https://brainly.com/question/10449635  [Answered by Calculista]
  • https://brainly.com/question/1596209  [Answered by AkhileshT]

Keywords :

Quadratic equation, standard form, completing squares method, Polynomial identities.