Solve -6x=-4x^2+4

$$- 6 x = - 4 x ^ { 2 } + 4$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

$$x=2$$

Steps Using Factoring By Grouping

Add 4x^{2} to both sides.

$$-6x+4x^{2}=4$$

Subtract 4 from both sides.

$$-6x+4x^{2}-4=0$$

Divide both sides by 2.

$$-3x+2x^{2}-2=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$2x^{2}-3x-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=2\left(-2\right)=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=-4 b=1$$

Rewrite 2x^{2}-3x-2 as \left(2x^{2}-4x\right)+\left(x-2\right).

$$\left(2x^{2}-4x\right)+\left(x-2\right)$$

Factor out 2x in 2x^{2}-4x.

$$2x\left(x-2\right)+x-2$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(2x+1\right)$$

To find equation solutions, solve x-2=0 and 2x+1=0.

$$x=2 x=-\frac{1}{2}$$

Steps Using the Quadratic Formula

Add 4x^{2} to both sides.

$$-6x+4x^{2}=4$$

Subtract 4 from both sides.

$$-6x+4x^{2}-4=0$$

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.

$$4x^{2}-6x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -6 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 4\left(-4\right)}}{2\times 4}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 4\left(-4\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-6\right)±\sqrt{36-16\left(-4\right)}}{2\times 4}$$

Multiply -16 times -4.

$$x=\frac{-\left(-6\right)±\sqrt{36+64}}{2\times 4}$$

Add 36 to 64.

$$x=\frac{-\left(-6\right)±\sqrt{100}}{2\times 4}$$

Take the square root of 100.

$$x=\frac{-\left(-6\right)±10}{2\times 4}$$

The opposite of -6 is 6.

$$x=\frac{6±10}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{6±10}{8}$$

Now solve the equation x=\frac{6±10}{8} when ± is plus. Add 6 to 10.

$$x=\frac{16}{8}$$

Divide 16 by 8.

$$x=2$$

Now solve the equation x=\frac{6±10}{8} when ± is minus. Subtract 10 from 6.

$$x=-\frac{4}{8}$$

Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{1}{2}$$

The equation is now solved.

$$x=2 x=-\frac{1}{2}$$

Steps for Completing the Square

Add 4x^{2} to both sides.

$$-6x+4x^{2}=4$$

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.

$$4x^{2}-6x=4$$

Divide both sides by 4.

$$\frac{4x^{2}-6x}{4}=\frac{4}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{6}{4}\right)x=\frac{4}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{3}{2}x=\frac{4}{4}$$

Divide 4 by 4.

$$x^{2}-\frac{3}{2}x=1$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$$

Add 1 to \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{25}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{5}{4} x-\frac{3}{4}=-\frac{5}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=2 x=-\frac{1}{2}$$