Solve (5x+2)(2x-3)-4((5x+23)(2x-3)/2)=0

$$( 5 x + 2 ) ( 2 x - 3 ) - 4 ( \frac { ( 5 x + 23 ) ( 2 x - 3 ) } { 2 } ) = 0$$

Solve for x

$$x = -\frac{44}{5} = -8\frac{4}{5} = -8.8$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 2.

$$2\left(5x+2\right)\left(2x-3\right)-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply 2 by 5x+2.

$$\left(10x+4\right)\left(2x-3\right)-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply 10x+4 by 2x-3 and combine like terms.

$$20x^{2}-22x-12-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply -4 by 5x+23.

$$20x^{2}-22x-12+\left(-20x-92\right)\left(2x-3\right)=0$$

Use the distributive property to multiply -20x-92 by 2x-3 and combine like terms.

$$20x^{2}-22x-12-40x^{2}-124x+276=0$$

Combine 20x^{2} and -40x^{2} to get -20x^{2}.

$$-20x^{2}-22x-12-124x+276=0$$

Combine -22x and -124x to get -146x.

$$-20x^{2}-146x-12+276=0$$

Add -12 and 276 to get 264.

$$-20x^{2}-146x+264=0$$

Divide both sides by 2.

$$-10x^{2}-73x+132=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -10x^{2}+ax+bx+132. To find a and b, set up a system to be solved.

$$a+b=-73 ab=-10\times 132=-1320$$

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 -1320.

$$1,-1320 2,-660 3,-440 4,-330 5,-264 6,-220 8,-165 10,-132 11,-120 12,-110 15,-88 20,-66 22,-60 24,-55 30,-44 33,-40$$

Calculate the sum for each pair.

$$1-1320=-1319 2-660=-658 3-440=-437 4-330=-326 5-264=-259 6-220=-214 8-165=-157 10-132=-122 11-120=-109 12-110=-98 15-88=-73 20-66=-46 22-60=-38 24-55=-31 30-44=-14 33-40=-7$$

The solution is the pair that gives sum -73.

$$a=15 b=-88$$

Rewrite -10x^{2}-73x+132 as \left(-10x^{2}+15x\right)+\left(-88x+132\right).

$$\left(-10x^{2}+15x\right)+\left(-88x+132\right)$$

Factor out -5x in the first and -44 in the second group.

$$-5x\left(2x-3\right)-44\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(-5x-44\right)$$

To find equation solutions, solve 2x-3=0 and -5x-44=0.

$$x=\frac{3}{2} x=-\frac{44}{5}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\left(5x+2\right)\left(2x-3\right)-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply 2 by 5x+2.

$$\left(10x+4\right)\left(2x-3\right)-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply 10x+4 by 2x-3 and combine like terms.

$$20x^{2}-22x-12-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply -4 by 5x+23.

$$20x^{2}-22x-12+\left(-20x-92\right)\left(2x-3\right)=0$$

Use the distributive property to multiply -20x-92 by 2x-3 and combine like terms.

$$20x^{2}-22x-12-40x^{2}-124x+276=0$$

Combine 20x^{2} and -40x^{2} to get -20x^{2}.

$$-20x^{2}-22x-12-124x+276=0$$

Combine -22x and -124x to get -146x.

$$-20x^{2}-146x-12+276=0$$

Add -12 and 276 to get 264.

$$-20x^{2}-146x+264=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, -146 for b, and 264 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-146\right)±\sqrt{\left(-146\right)^{2}-4\left(-20\right)\times 264}}{2\left(-20\right)}$$

Square -146.

$$x=\frac{-\left(-146\right)±\sqrt{21316-4\left(-20\right)\times 264}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-\left(-146\right)±\sqrt{21316+80\times 264}}{2\left(-20\right)}$$

Multiply 80 times 264.

$$x=\frac{-\left(-146\right)±\sqrt{21316+21120}}{2\left(-20\right)}$$

Add 21316 to 21120.

$$x=\frac{-\left(-146\right)±\sqrt{42436}}{2\left(-20\right)}$$

Take the square root of 42436.

$$x=\frac{-\left(-146\right)±206}{2\left(-20\right)}$$

The opposite of -146 is 146.

$$x=\frac{146±206}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{146±206}{-40}$$

Now solve the equation x=\frac{146±206}{-40} when ± is plus. Add 146 to 206.

$$x=\frac{352}{-40}$$

Reduce the fraction \frac{352}{-40} to lowest terms by extracting and canceling out 8.

$$x=-\frac{44}{5}$$

Now solve the equation x=\frac{146±206}{-40} when ± is minus. Subtract 206 from 146.

$$x=-\frac{60}{-40}$$

Reduce the fraction \frac{-60}{-40} to lowest terms by extracting and canceling out 20.

$$x=\frac{3}{2}$$

The equation is now solved.

$$x=-\frac{44}{5} x=\frac{3}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2\left(5x+2\right)\left(2x-3\right)-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply 2 by 5x+2.

$$\left(10x+4\right)\left(2x-3\right)-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply 10x+4 by 2x-3 and combine like terms.

$$20x^{2}-22x-12-4\left(5x+23\right)\left(2x-3\right)=0$$

Use the distributive property to multiply -4 by 5x+23.

$$20x^{2}-22x-12+\left(-20x-92\right)\left(2x-3\right)=0$$

Use the distributive property to multiply -20x-92 by 2x-3 and combine like terms.

$$20x^{2}-22x-12-40x^{2}-124x+276=0$$

Combine 20x^{2} and -40x^{2} to get -20x^{2}.

$$-20x^{2}-22x-12-124x+276=0$$

Combine -22x and -124x to get -146x.

$$-20x^{2}-146x-12+276=0$$

Add -12 and 276 to get 264.

$$-20x^{2}-146x+264=0$$

Subtract 264 from both sides. Anything subtracted from zero gives its negation.

$$-20x^{2}-146x=-264$$

Divide both sides by -20.

$$\frac{-20x^{2}-146x}{-20}=-\frac{264}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\left(-\frac{146}{-20}\right)x=-\frac{264}{-20}$$

Reduce the fraction \frac{-146}{-20} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{73}{10}x=-\frac{264}{-20}$$

Reduce the fraction \frac{-264}{-20} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{73}{10}x=\frac{66}{5}$$

Divide \frac{73}{10}, the coefficient of the x term, by 2 to get \frac{73}{20}. Then add the square of \frac{73}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{73}{10}x+\left(\frac{73}{20}\right)^{2}=\frac{66}{5}+\left(\frac{73}{20}\right)^{2}$$

Square \frac{73}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{73}{10}x+\frac{5329}{400}=\frac{66}{5}+\frac{5329}{400}$$

Add \frac{66}{5} to \frac{5329}{400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{73}{10}x+\frac{5329}{400}=\frac{10609}{400}$$

Factor x^{2}+\frac{73}{10}x+\frac{5329}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{73}{20}\right)^{2}=\frac{10609}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{73}{20}\right)^{2}}=\sqrt{\frac{10609}{400}}$$

Simplify.

$$x+\frac{73}{20}=\frac{103}{20} x+\frac{73}{20}=-\frac{103}{20}$$

Subtract \frac{73}{20} from both sides of the equation.

$$x=\frac{3}{2} x=-\frac{44}{5}$$