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4x+4x^{2}\times \frac{1}{2}=6
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x^{2}, the least common multiple of x,2,4x^{2}.
4x+2x^{2}=6
Multiply 4 and \frac{1}{2} to get 2.
4x+2x^{2}-6=0
Subtract 6 from both sides.
2x+x^{2}-3=0
Divide both sides by 2.
x^{2}+2x-3=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=2 ab=1\left(-3\right)=-3
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-3. To find a and b, set up a system to be solved.
a=-1 b=3
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.
\left(x^{2}-x\right)+\left(3x-3\right)
Rewrite x^{2}+2x-3 as \left(x^{2}-x\right)+\left(3x-3\right).
x\left(x-1\right)+3\left(x-1\right)
Factor out x in the first and 3 in the second group.
\left(x-1\right)\left(x+3\right)
Factor out common term x-1 by using distributive property.
x=1 x=-3
To find equation solutions, solve x-1=0 and x+3=0.
4x+4x^{2}\times \frac{1}{2}=6
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x^{2}, the least common multiple of x,2,4x^{2}.
4x+2x^{2}=6
Multiply 4 and \frac{1}{2} to get 2.
4x+2x^{2}-6=0
Subtract 6 from both sides.
2x^{2}+4x-6=0
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-6\right)}}{2\times 2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\times 2\left(-6\right)}}{2\times 2}
Square 4.
x=\frac{-4±\sqrt{16-8\left(-6\right)}}{2\times 2}
Multiply -4 times 2.
x=\frac{-4±\sqrt{16+48}}{2\times 2}
Multiply -8 times -6.
x=\frac{-4±\sqrt{64}}{2\times 2}
Add 16 to 48.
x=\frac{-4±8}{2\times 2}
Take the square root of 64.
x=\frac{-4±8}{4}
Multiply 2 times 2.
x=\frac{4}{4}
Now solve the equation x=\frac{-4±8}{4} when ± is plus. Add -4 to 8.
x=1
Divide 4 by 4.
x=-\frac{12}{4}
Now solve the equation x=\frac{-4±8}{4} when ± is minus. Subtract 8 from -4.
x=-3
Divide -12 by 4.
x=1 x=-3
The equation is now solved.
4x+4x^{2}\times \frac{1}{2}=6
Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x^{2}, the least common multiple of x,2,4x^{2}.
4x+2x^{2}=6
Multiply 4 and \frac{1}{2} to get 2.
2x^{2}+4x=6
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
\frac{2x^{2}+4x}{2}=\frac{6}{2}
Divide both sides by 2.
x^{2}+\frac{4}{2}x=\frac{6}{2}
Dividing by 2 undoes the multiplication by 2.
x^{2}+2x=\frac{6}{2}
Divide 4 by 2.
x^{2}+2x=3
Divide 6 by 2.
x^{2}+2x+1^{2}=3+1^{2}
Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+2x+1=3+1
Square 1.
x^{2}+2x+1=4
Add 3 to 1.
\left(x+1\right)^{2}=4
Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x+1\right)^{2}}=\sqrt{4}
Take the square root of both sides of the equation.
x+1=2 x+1=-2
Simplify.
x=1 x=-3
Subtract 1 from both sides of the equation.