Solve left(4xright)^2-4x-6=0

$${ \left(4x \right) }^{ 2 } -4x-6 = 0$$

Solve for x

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

$$x=\frac{3}{4}=0.75$$

Steps Using Factoring By Grouping

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-4x-6=0$$

Divide both sides by 2.

$$8x^{2}-2x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=8\left(-3\right)=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite 8x^{2}-2x-3 as \left(8x^{2}-6x\right)+\left(4x-3\right).

$$\left(8x^{2}-6x\right)+\left(4x-3\right)$$

Factor out 2x in 8x^{2}-6x.

$$2x\left(4x-3\right)+4x-3$$

Factor out common term 4x-3 by using distributive property.

$$\left(4x-3\right)\left(2x+1\right)$$

To find equation solutions, solve 4x-3=0 and 2x+1=0.

$$x=\frac{3}{4} x=-\frac{1}{2}$$

Steps Using the Quadratic Formula

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-4x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -4 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

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

Multiply -4 times 16.

$$x=\frac{-\left(-4\right)±\sqrt{16-64\left(-6\right)}}{2\times 16}$$

Multiply -64 times -6.

$$x=\frac{-\left(-4\right)±\sqrt{16+384}}{2\times 16}$$

Add 16 to 384.

$$x=\frac{-\left(-4\right)±\sqrt{400}}{2\times 16}$$

Take the square root of 400.

$$x=\frac{-\left(-4\right)±20}{2\times 16}$$

The opposite of -4 is 4.

$$x=\frac{4±20}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{4±20}{32}$$

Now solve the equation x=\frac{4±20}{32} when ± is plus. Add 4 to 20.

$$x=\frac{24}{32}$$

Reduce the fraction \frac{24}{32} to lowest terms by extracting and canceling out 8.

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

Now solve the equation x=\frac{4±20}{32} when ± is minus. Subtract 20 from 4.

$$x=-\frac{16}{32}$$

Reduce the fraction \frac{-16}{32} to lowest terms by extracting and canceling out 16.

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

The equation is now solved.

$$x=\frac{3}{4} x=-\frac{1}{2}$$

Steps for Completing the Square

Expand \left(4x\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-4x-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$16x^{2}-4x=6$$

Divide both sides by 16.

$$\frac{16x^{2}-4x}{16}=\frac{6}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{4}{16}\right)x=\frac{6}{16}$$

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

$$x^{2}-\frac{1}{4}x=\frac{6}{16}$$

Reduce the fraction \frac{6}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{1}{4}x=\frac{3}{8}$$

Divide -\frac{1}{4}, the coefficient of the x term, by 2 to get -\frac{1}{8}. Then add the square of -\frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{3}{8}+\frac{1}{64}$$

Add \frac{3}{8} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{25}{64}$$

Factor x^{2}-\frac{1}{4}x+\frac{1}{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}.

$$\left(x-\frac{1}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$x-\frac{1}{8}=\frac{5}{8} x-\frac{1}{8}=-\frac{5}{8}$$

Add \frac{1}{8} to both sides of the equation.

$$x=\frac{3}{4} x=-\frac{1}{2}$$