Solve 42x(4x-1)=18

$$42 x ( 4 x - 1 ) = 18$$

Solve for x

$$x=\frac{\sqrt{385}}{56}+\frac{1}{8}\approx 0.475382444$$

$$x=-\frac{\sqrt{385}}{56}+\frac{1}{8}\approx -0.225382444$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 42x by 4x-1.

$$168x^{2}-42x=18$$

Subtract 18 from both sides.

$$168x^{2}-42x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 168 for a, -42 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-42\right)±\sqrt{\left(-42\right)^{2}-4\times 168\left(-18\right)}}{2\times 168}$$

Square -42.

$$x=\frac{-\left(-42\right)±\sqrt{1764-4\times 168\left(-18\right)}}{2\times 168}$$

Multiply -4 times 168.

$$x=\frac{-\left(-42\right)±\sqrt{1764-672\left(-18\right)}}{2\times 168}$$

Multiply -672 times -18.

$$x=\frac{-\left(-42\right)±\sqrt{1764+12096}}{2\times 168}$$

Add 1764 to 12096.

$$x=\frac{-\left(-42\right)±\sqrt{13860}}{2\times 168}$$

Take the square root of 13860.

$$x=\frac{-\left(-42\right)±6\sqrt{385}}{2\times 168}$$

The opposite of -42 is 42.

$$x=\frac{42±6\sqrt{385}}{2\times 168}$$

Multiply 2 times 168.

$$x=\frac{42±6\sqrt{385}}{336}$$

Now solve the equation x=\frac{42±6\sqrt{385}}{336} when ± is plus. Add 42 to 6\sqrt{385}.

$$x=\frac{6\sqrt{385}+42}{336}$$

Divide 42+6\sqrt{385} by 336.

$$x=\frac{\sqrt{385}}{56}+\frac{1}{8}$$

Now solve the equation x=\frac{42±6\sqrt{385}}{336} when ± is minus. Subtract 6\sqrt{385} from 42.

$$x=\frac{42-6\sqrt{385}}{336}$$

Divide 42-6\sqrt{385} by 336.

$$x=-\frac{\sqrt{385}}{56}+\frac{1}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{385}}{56}+\frac{1}{8} x=-\frac{\sqrt{385}}{56}+\frac{1}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 42x by 4x-1.

$$168x^{2}-42x=18$$

Divide both sides by 168.

$$\frac{168x^{2}-42x}{168}=\frac{18}{168}$$

Dividing by 168 undoes the multiplication by 168.

$$x^{2}+\left(-\frac{42}{168}\right)x=\frac{18}{168}$$

Reduce the fraction \frac{-42}{168} to lowest terms by extracting and canceling out 42.

$$x^{2}-\frac{1}{4}x=\frac{18}{168}$$

Reduce the fraction \frac{18}{168} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{1}{4}x=\frac{3}{28}$$

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.

$$x^{2}-\frac{1}{4}x+\left(-\frac{1}{8}\right)^{2}=\frac{3}{28}+\left(-\frac{1}{8}\right)^{2}$$

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

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{3}{28}+\frac{1}{64}$$

Add \frac{3}{28} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{55}{448}$$

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}.

$$\left(x-\frac{1}{8}\right)^{2}=\frac{55}{448}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{8}\right)^{2}}=\sqrt{\frac{55}{448}}$$

Simplify.

$$x-\frac{1}{8}=\frac{\sqrt{385}}{56} x-\frac{1}{8}=-\frac{\sqrt{385}}{56}$$

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

$$x=\frac{\sqrt{385}}{56}+\frac{1}{8} x=-\frac{\sqrt{385}}{56}+\frac{1}{8}$$