Solve 6x=-8x^2+2

$$6 x = - 8 x ^ { 2 } + 2$$

Solve for x

$$x=-1$$

$$x=\frac{1}{4}=0.25$$

Steps Using Factoring By Grouping

Add 8x^{2} to both sides.

$$6x+8x^{2}=2$$

Subtract 2 from both sides.

$$6x+8x^{2}-2=0$$

Divide both sides by 2.

$$3x+4x^{2}-1=0$$

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

$$4x^{2}+3x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=3 ab=4\left(-1\right)=-4$$

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. 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.

$$a=-1 b=4$$

Rewrite 4x^{2}+3x-1 as \left(4x^{2}-x\right)+\left(4x-1\right).

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

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

$$x\left(4x-1\right)+4x-1$$

Factor out common term 4x-1 by using distributive property.

$$\left(4x-1\right)\left(x+1\right)$$

To find equation solutions, solve 4x-1=0 and x+1=0.

$$x=\frac{1}{4} x=-1$$

Steps Using the Quadratic Formula

Add 8x^{2} to both sides.

$$6x+8x^{2}=2$$

Subtract 2 from both sides.

$$6x+8x^{2}-2=0$$

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.

$$8x^{2}+6x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 6 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 8\left(-2\right)}}{2\times 8}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 8\left(-2\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-6±\sqrt{36-32\left(-2\right)}}{2\times 8}$$

Multiply -32 times -2.

$$x=\frac{-6±\sqrt{36+64}}{2\times 8}$$

Add 36 to 64.

$$x=\frac{-6±\sqrt{100}}{2\times 8}$$

Take the square root of 100.

$$x=\frac{-6±10}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-6±10}{16}$$

Now solve the equation x=\frac{-6±10}{16} when ± is plus. Add -6 to 10.

$$x=\frac{4}{16}$$

Reduce the fraction \frac{4}{16} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{4}$$

Now solve the equation x=\frac{-6±10}{16} when ± is minus. Subtract 10 from -6.

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

Divide -16 by 16.

$$x=-1$$

The equation is now solved.

$$x=\frac{1}{4} x=-1$$

Steps for Completing the Square

Add 8x^{2} to both sides.

$$6x+8x^{2}=2$$

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.

$$8x^{2}+6x=2$$

Divide both sides by 8.

$$\frac{8x^{2}+6x}{8}=\frac{2}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{6}{8}x=\frac{2}{8}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{3}{4}x=\frac{2}{8}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{3}{4}x=\frac{1}{4}$$

Divide \frac{3}{4}, the coefficient of the x term, by 2 to get \frac{3}{8}. Then add the square of \frac{3}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=\frac{1}{4}+\left(\frac{3}{8}\right)^{2}$$

Square \frac{3}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{1}{4}+\frac{9}{64}$$

Add \frac{1}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{25}{64}$$

Factor x^{2}+\frac{3}{4}x+\frac{9}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$x+\frac{3}{8}=\frac{5}{8} x+\frac{3}{8}=-\frac{5}{8}$$

Subtract \frac{3}{8} from both sides of the equation.

$$x=\frac{1}{4} x=-1$$