Solve 2/3x^2-x+3/2=3

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

Solve for x

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

$$x=3$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 6, the least common multiple of 3,2.

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

Use the distributive property to multiply -3 by x+3.

$$4x^{2}-3x-9=18$$

Subtract 18 from both sides.

$$4x^{2}-3x-9-18=0$$

Subtract 18 from -9 to get -27.

$$4x^{2}-3x-27=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-27. To find a and b, set up a system to be solved.

$$a+b=-3 ab=4\left(-27\right)=-108$$

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 -108.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Calculate the sum for each pair.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

The solution is the pair that gives sum -3.

$$a=-12 b=9$$

Rewrite 4x^{2}-3x-27 as \left(4x^{2}-12x\right)+\left(9x-27\right).

$$\left(4x^{2}-12x\right)+\left(9x-27\right)$$

Factor out 4x in the first and 9 in the second group.

$$4x\left(x-3\right)+9\left(x-3\right)$$

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

$$\left(x-3\right)\left(4x+9\right)$$

To find equation solutions, solve x-3=0 and 4x+9=0.

$$x=3 x=-\frac{9}{4}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,2.

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

Use the distributive property to multiply -3 by x+3.

$$4x^{2}-3x-9=18$$

Subtract 18 from both sides.

$$4x^{2}-3x-9-18=0$$

Subtract 18 from -9 to get -27.

$$4x^{2}-3x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -3 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 4\left(-27\right)}}{2\times 4}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 4\left(-27\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-3\right)±\sqrt{9-16\left(-27\right)}}{2\times 4}$$

Multiply -16 times -27.

$$x=\frac{-\left(-3\right)±\sqrt{9+432}}{2\times 4}$$

Add 9 to 432.

$$x=\frac{-\left(-3\right)±\sqrt{441}}{2\times 4}$$

Take the square root of 441.

$$x=\frac{-\left(-3\right)±21}{2\times 4}$$

The opposite of -3 is 3.

$$x=\frac{3±21}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{3±21}{8}$$

Now solve the equation x=\frac{3±21}{8} when ± is plus. Add 3 to 21.

$$x=\frac{24}{8}$$

Divide 24 by 8.

$$x=3$$

Now solve the equation x=\frac{3±21}{8} when ± is minus. Subtract 21 from 3.

$$x=-\frac{18}{8}$$

Reduce the fraction \frac{-18}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{9}{4}$$

The equation is now solved.

$$x=3 x=-\frac{9}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 3,2.

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

Use the distributive property to multiply -3 by x+3.

$$4x^{2}-3x-9=18$$

Add 9 to both sides.

$$4x^{2}-3x=18+9$$

Add 18 and 9 to get 27.

$$4x^{2}-3x=27$$

Divide both sides by 4.

$$\frac{4x^{2}-3x}{4}=\frac{27}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{3}{4}x=\frac{27}{4}$$

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.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=\frac{27}{4}+\left(-\frac{3}{8}\right)^{2}$$

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

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{27}{4}+\frac{9}{64}$$

Add \frac{27}{4} to \frac{9}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=\frac{441}{64}$$

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}.

$$\left(x-\frac{3}{8}\right)^{2}=\frac{441}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{\frac{441}{64}}$$

Simplify.

$$x-\frac{3}{8}=\frac{21}{8} x-\frac{3}{8}=-\frac{21}{8}$$

Add \frac{3}{8} to both sides of the equation.

$$x=3 x=-\frac{9}{4}$$