Solve -18x^2=-27x+4

$$- 18 x ^ { 2 } = - 27 x + 4$$

Solve for x

$$x=\frac{1}{6}\approx 0.166666667$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Steps Using Factoring By Grouping

Add 27x to both sides.

$$-18x^{2}+27x=4$$

Subtract 4 from both sides.

$$-18x^{2}+27x-4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -18x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=27 ab=-18\left(-4\right)=72$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 27.

$$a=24 b=3$$

Rewrite -18x^{2}+27x-4 as \left(-18x^{2}+24x\right)+\left(3x-4\right).

$$\left(-18x^{2}+24x\right)+\left(3x-4\right)$$

Factor out -6x in -18x^{2}+24x.

$$-6x\left(3x-4\right)+3x-4$$

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

$$\left(3x-4\right)\left(-6x+1\right)$$

To find equation solutions, solve 3x-4=0 and -6x+1=0.

$$x=\frac{4}{3} x=\frac{1}{6}$$

Steps Using the Quadratic Formula

Add 27x to both sides.

$$-18x^{2}+27x=4$$

Subtract 4 from both sides.

$$-18x^{2}+27x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, 27 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-18\right)\left(-4\right)}}{2\left(-18\right)}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\left(-18\right)\left(-4\right)}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-27±\sqrt{729+72\left(-4\right)}}{2\left(-18\right)}$$

Multiply 72 times -4.

$$x=\frac{-27±\sqrt{729-288}}{2\left(-18\right)}$$

Add 729 to -288.

$$x=\frac{-27±\sqrt{441}}{2\left(-18\right)}$$

Take the square root of 441.

$$x=\frac{-27±21}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-27±21}{-36}$$

Now solve the equation x=\frac{-27±21}{-36} when ± is plus. Add -27 to 21.

$$x=-\frac{6}{-36}$$

Reduce the fraction \frac{-6}{-36} to lowest terms by extracting and canceling out 6.

$$x=\frac{1}{6}$$

Now solve the equation x=\frac{-27±21}{-36} when ± is minus. Subtract 21 from -27.

$$x=-\frac{48}{-36}$$

Reduce the fraction \frac{-48}{-36} to lowest terms by extracting and canceling out 12.

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

The equation is now solved.

$$x=\frac{1}{6} x=\frac{4}{3}$$

Steps for Completing the Square

Add 27x to both sides.

$$-18x^{2}+27x=4$$

Divide both sides by -18.

$$\frac{-18x^{2}+27x}{-18}=\frac{4}{-18}$$

Dividing by -18 undoes the multiplication by -18.

$$x^{2}+\frac{27}{-18}x=\frac{4}{-18}$$

Reduce the fraction \frac{27}{-18} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{3}{2}x=\frac{4}{-18}$$

Reduce the fraction \frac{4}{-18} to lowest terms by extracting and canceling out 2.

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

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.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{2}{9}+\left(-\frac{3}{4}\right)^{2}$$

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

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

Add -\frac{2}{9} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{49}{144}$$

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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$x-\frac{3}{4}=\frac{7}{12} x-\frac{3}{4}=-\frac{7}{12}$$

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

$$x=\frac{4}{3} x=\frac{1}{6}$$