need help completing the square

lil_Dk

New member
Help me by showin (in steps) me how to completing the square......

4g(square)+3g+1=0

galactus

Super Moderator
Staff member
You can use different techniques. That 4 as the leading coefficient is tripping you up, ain't it?. They'll do that.

$$\displaystyle \L\\4x^{2}+3x+1=0$$

$$\displaystyle \L\\4x^{2}+3x=-1$$

$$\displaystyle \L\\4(x^{2}+\frac{3}{4}x)=-1$$

Now, take one-half the coefficient of x, square it, and add it inside the

parentheses;

take 4 times the number you just found and add it to the right side(-1).

one-half the coefficient of x is 3/8. The square of 3/8 is 9/64.

4 times 9/64 is 9/16.

$$\displaystyle \L\\4(x^{2}+\frac{3}{4}x+\frac{9}{64})=-1+\frac{9}{16}$$

Can you finish up?.

stapel

Super Moderator
Staff member
Given ax<sup>2</sup> + bx + c = 0, the completing-the-square process is, step-by-step, as follows:

. . . . .Take the original equation:

. . . . .$$\displaystyle \large{ax^2\,+\,bx\,+\,c\,=\,0}$$

. . . . .Move the constant term to the other side:

. . . . .$$\displaystyle \large{ax^2\,+\,bx\,=\,-c}$$

. . . . .Factor out whatever is multiplied on the squared term:

. . . . .$$\displaystyle \large{a\left(x^2\,+\,\frac{b}{a}x\right)\,=\,-c}$$

. . . . .Take one-half of the coefficient of the linear term,
. . . . .square it, and add on both sides, taking account of
. . . . .anything you may have factored out:

. . . . .$$\displaystyle \large{\left(\frac{1}{2}\right)\,\left(\frac{b}{a}\right)\. =\, \frac{b}{2a}}$$

. . . . .$$\displaystyle \large{\left(\frac{b}{2a}\right)^2\, =\, \frac{b^2}{4a^2}}$$

. . . . .$$\displaystyle \large{a\left(x^2\,+\,\frac{b}{a}x\,+\,\frac{b}{4a^2}\right)\,=\,a\left(\frac{b}{4a^2}\right)\,-\,c \,=\,\frac{b}{4a}\,-\,c}$$

. . . . .Complete the square on the left-hand side:

. . . . .$$\displaystyle \large{a\left(x\,+\,\frac{b}{2a}\right)^2\,=\,\frac{b}{4a}\,-\,c}$$

. . . . .Divide through by whatever was factored out:

. . . . .$$\displaystyle \large{(x\,+\,\frac{b}{2a})^2\,=\,\frac{b}{4a^2}\,-\,\frac{c}{a}}$$

. . . . .Take the square root:

. . . . .$$\displaystyle \large{x\,+\,\frac{b}{2a}\,=\,\pm\,\sqrt{\frac{b}{4a^2}\,-\,\frac{c}{a}}}$$

. . . . .Move whatever is with the variable over onto the other side:

. . . . .$$\displaystyle \large{x\,= \,-\frac{b}{2a}\,\pm\,\sqrt{\frac{b}{4a^2}\,-\,\frac{c}{a}}}$$

So, for instance:

. . . . .$$\displaystyle \large{3x^2\, +\, 5x\, -\, 4\, =\, 0}$$

. . . . .$$\displaystyle \large{3x^2\,+\,5x\,=\,4}$$

. . . . .$$\displaystyle \large{3(x^2\,+\,\frac{5}{3}x)\,= \,4}$$

. . . . .$$\displaystyle \large{3(x^2\,+\,\frac{5}{3}x\,+\,\frac{25}{36})\, =\,4\,+\,3\left(\frac{25}{36}\right)\, =\,4\,+\,\frac{25}{12}\,=\,\frac{73}{12}}$$

. . . . .$$\displaystyle \large{3(x\,+\,\frac{5}{6})^2\,=\,\frac{73}{12}}$$

. . . . .$$\displaystyle \large{(x\,+\,\frac{5}{6})^2\,=\,\frac{73}{36}}$$

. . . . .$$\displaystyle \large{x\,+\,\frac{5}{6}\,= \,\pm\,\sqrt{\frac{73}{36}}}$$

. . . . .$$\displaystyle \large{x\,= \,-\frac{5}{6}\,\pm\,\sqrt{\frac{73}{36}}}$$

Simplify a bit, and you're done.

Your example works the same way.

Eliz.

Edit: Corrected typoes graciously pointed out by galactus. Thank you!

soroban

Elite Member
Hello, lil_Dk!

This quadratic has complex roots; I assume you're cool with that . . .

$$\displaystyle \L4g^2\,+\,3g\,+\,1\:=\:0$$
I prefer to get rid of the leading coefficient first . . .

Divide by 4: $$\displaystyle \L\,g^2\,+\,\frac{3}{4}g\,+\,\frac{1}{4}\:=\:0$$

"Move" the constant term: $$\displaystyle \L\,g^2\,+\,\frac{3}{4}g\;=\;-\frac{1}{4}$$

Complete the square:
$$\displaystyle \;\;$$Take one-half of $$\displaystyle \frac{3}{4}$$ and square: $$\displaystyle \,\left(\frac{1}{2}\cdot\frac{3}{4}\right)^2\,=\,\frac{9}{64}$$

Add to both sides: $$\displaystyle \L\,g^2\,+\,\frac{3}{4}g\,+\,\frac{9}{64}\;=\;-\frac{1}{4}\,+\,\frac{9}{64}$$

Simplify: $$\displaystyle \L\,\left(g\,+\,\frac{3}{8}\right)^2\:=\: -\frac{7}{64}$$

Take square roots: $$\displaystyle \L\,g\,+\,\frac{3}{8}\:=\:\pm\sqrt{-\frac{7}{64}}\:=\:\pm\frac{i\sqrt{7}}{8}$$

Therefore: $$\displaystyle \L\,g\:=\:-\frac{3}{8}\,\pm\,\frac{i\sqrt{7}}{8}\:=\:\frac{-3\,\pm\,i\sqrt{7}}{8}$$