"Complex squares, what are the steps: 2x2−x(5+m)+(3+m)=0" was answered by our community

High School
Algebra
Algebra I
Quadratic function and equation

Anika Klein

Answered question

2022-01-21

Complex squares, what are the steps:

2 x^{2} - x (5 + m) + (3 + m) = 0

Answer & Explanation

ma90t66690

Beginner2022-01-22Added 7 answers

Step 1
To complete the square on the left hand side of your equation, first factor out the 2:
1) $2 x^{2} - x (5 + m) + (3 + m)$
$= 2 (x^{2} - \frac{5 + m}{2} + \frac{3 + m}{2})$
Write down what you want:
2) $x^{2} - \frac{5 + m}{2} x + \frac{3 + m}{2}$
$= {(x - h)}^{2} + k$
Write the right hand side of (2) as
3) $x^{2} - 2 h x + h^{2} + k$
Looking at the x term of the left hand side of (2) with the x term of (3), write down an equation and solve for h:
$- \frac{5 + m}{2} x = - 2 h x \Rightarrow h = \frac{5 + m}{4}$
Now use equations (2) and (3) to find the value of k:
$\frac{3 + m}{2} = h^{2} + k \Rightarrow k = \frac{3 + m}{2} - h^{2} \Rightarrow k$
$= \frac{3 + m}{2} - {(\frac{5 + m}{4})}^{2}$
So
$x^{2} - \frac{5 + m}{2} x + \frac{3 + m}{2}$
$= {(x - h)}^{2} + k$
$= {(x - \frac{5 + m}{4})}^{2} + \frac{3 + m}{2} - {(\frac{5 + m}{4})}^{2}$
and
$2 x^{2} - x (5 + m) + (3 + m)$
$= 2 (x^{2} - \frac{5 + m}{2} x + \frac{3 + m}{2})$
$= 2 ((x - \frac{5 + m}{4})^{2} + \frac{3 + m}{2} - (\frac{5 + m}{4})^{2})$
$= 2 {(x - \frac{5 + m}{4})}^{2} + 3 + m - 2 {(\frac{5 + m}{4})}^{2}$

Sean Becker

Beginner2022-01-23Added 16 answers

Step 1

x^{2} - \frac{(5 + m)}{2} x + \frac{3 + m}{2} = 0

x^{2} - 2 \frac{(5 + m)}{4} x + \frac{3 + m}{2} = 0

x^{2} - 2 (\frac{5 + m}{4}) x + {(\frac{5 + m}{4})}^{2} - {(\frac{5 + m}{4})}^{2} + \frac{3 + m}{2} = 0

{(x - \frac{5 + m}{4})}^{2} - (\frac{{(5 + m)}^{2}}{16} - \frac{3 + m}{2}) = 0

RizerMix

Expert2022-01-27Added 656 answers

In general:

a x^{2} + b x + c

a (x + \frac{b}{2 a})^{2} + c - \frac{b^{2}}{4 a}

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