Solve (x+3/2)(x-3/2)-8x(x+1)=4(2x-3/2)

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

Solve for x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x=\frac{3}{14}\approx 0.214285714$$

Steps Using the Quadratic Formula

Consider \left(x+\frac{3}{2}\right)\left(x-\frac{3}{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square \frac{3}{2}.

$$x^{2}-\frac{9}{4}-8x\left(x+1\right)=4\left(2x-\frac{3}{2}\right)$$

Use the distributive property to multiply 4 by 2x-\frac{3}{2}.

$$x^{2}-\frac{9}{4}-8x\left(x+1\right)=8x-6$$

Subtract 8x from both sides.

$$x^{2}-\frac{9}{4}-8x\left(x+1\right)-8x=-6$$

Add 6 to both sides.

$$x^{2}-\frac{9}{4}-8x\left(x+1\right)-8x+6=0$$

Use the distributive property to multiply -8x by x+1.

$$x^{2}-\frac{9}{4}-8x^{2}-8x-8x+6=0$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}-\frac{9}{4}-8x-8x+6=0$$

Combine -8x and -8x to get -16x.

$$-7x^{2}-\frac{9}{4}-16x+6=0$$

Add -\frac{9}{4} and 6 to get \frac{15}{4}.

$$-7x^{2}+\frac{15}{4}-16x=0$$

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.

$$-7x^{2}-16x+\frac{15}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, -16 for b, and \frac{15}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-7\right)\times \frac{15}{4}}}{2\left(-7\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-7\right)\times \frac{15}{4}}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-\left(-16\right)±\sqrt{256+28\times \frac{15}{4}}}{2\left(-7\right)}$$

Multiply 28 times \frac{15}{4}.

$$x=\frac{-\left(-16\right)±\sqrt{256+105}}{2\left(-7\right)}$$

Add 256 to 105.

$$x=\frac{-\left(-16\right)±\sqrt{361}}{2\left(-7\right)}$$

Take the square root of 361.

$$x=\frac{-\left(-16\right)±19}{2\left(-7\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±19}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{16±19}{-14}$$

Now solve the equation x=\frac{16±19}{-14} when ± is plus. Add 16 to 19.

$$x=\frac{35}{-14}$$

Reduce the fraction \frac{35}{-14} to lowest terms by extracting and canceling out 7.

$$x=-\frac{5}{2}$$

Now solve the equation x=\frac{16±19}{-14} when ± is minus. Subtract 19 from 16.

$$x=-\frac{3}{-14}$$

Divide -3 by -14.

$$x=\frac{3}{14}$$

The equation is now solved.

$$x=-\frac{5}{2} x=\frac{3}{14}$$

Steps for Completing the Square

Consider \left(x+\frac{3}{2}\right)\left(x-\frac{3}{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square \frac{3}{2}.

$$x^{2}-\frac{9}{4}-8x\left(x+1\right)=4\left(2x-\frac{3}{2}\right)$$

Use the distributive property to multiply 4 by 2x-\frac{3}{2}.

$$x^{2}-\frac{9}{4}-8x\left(x+1\right)=8x-6$$

Subtract 8x from both sides.

$$x^{2}-\frac{9}{4}-8x\left(x+1\right)-8x=-6$$

Use the distributive property to multiply -8x by x+1.

$$x^{2}-\frac{9}{4}-8x^{2}-8x-8x=-6$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}-\frac{9}{4}-8x-8x=-6$$

Combine -8x and -8x to get -16x.

$$-7x^{2}-\frac{9}{4}-16x=-6$$

Add \frac{9}{4} to both sides.

$$-7x^{2}-16x=-6+\frac{9}{4}$$

Add -6 and \frac{9}{4} to get -\frac{15}{4}.

$$-7x^{2}-16x=-\frac{15}{4}$$

Divide both sides by -7.

$$\frac{-7x^{2}-16x}{-7}=-\frac{\frac{15}{4}}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\left(-\frac{16}{-7}\right)x=-\frac{\frac{15}{4}}{-7}$$

Divide -16 by -7.

$$x^{2}+\frac{16}{7}x=-\frac{\frac{15}{4}}{-7}$$

Divide -\frac{15}{4} by -7.

$$x^{2}+\frac{16}{7}x=\frac{15}{28}$$

Divide \frac{16}{7}, the coefficient of the x term, by 2 to get \frac{8}{7}. Then add the square of \frac{8}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{7}x+\left(\frac{8}{7}\right)^{2}=\frac{15}{28}+\left(\frac{8}{7}\right)^{2}$$

Square \frac{8}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=\frac{15}{28}+\frac{64}{49}$$

Add \frac{15}{28} to \frac{64}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=\frac{361}{196}$$

Factor x^{2}+\frac{16}{7}x+\frac{64}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{7}\right)^{2}=\frac{361}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{7}\right)^{2}}=\sqrt{\frac{361}{196}}$$

Simplify.

$$x+\frac{8}{7}=\frac{19}{14} x+\frac{8}{7}=-\frac{19}{14}$$

Subtract \frac{8}{7} from both sides of the equation.

$$x=\frac{3}{14} x=-\frac{5}{2}$$