Solve x(2x+9)-4x-18=0

$$x ( 2 x + 9 ) - 4 x - 18 = 0$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=2$$

Steps Using Factoring By Grouping

Use the distributive property to multiply x by 2x+9.

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

Combine 9x and -4x to get 5x.

$$2x^{2}+5x-18=0$$

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

$$a+b=5 ab=2\left(-18\right)=-36$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -36.

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

Calculate the sum for each pair.

$$-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0$$

The solution is the pair that gives sum 5.

$$a=-4 b=9$$

Rewrite 2x^{2}+5x-18 as \left(2x^{2}-4x\right)+\left(9x-18\right).

$$\left(2x^{2}-4x\right)+\left(9x-18\right)$$

Factor out 2x in the first and 9 in the second group.

$$2x\left(x-2\right)+9\left(x-2\right)$$

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

$$\left(x-2\right)\left(2x+9\right)$$

To find equation solutions, solve x-2=0 and 2x+9=0.

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

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 2x+9.

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

Combine 9x and -4x to get 5x.

$$2x^{2}+5x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 5 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-18\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\left(-18\right)}}{2\times 2}$$

Multiply -8 times -18.

$$x=\frac{-5±\sqrt{25+144}}{2\times 2}$$

Add 25 to 144.

$$x=\frac{-5±\sqrt{169}}{2\times 2}$$

Take the square root of 169.

$$x=\frac{-5±13}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±13}{4}$$

Now solve the equation x=\frac{-5±13}{4} when ± is plus. Add -5 to 13.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{-5±13}{4} when ± is minus. Subtract 13 from -5.

$$x=-\frac{18}{4}$$

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

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply x by 2x+9.

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

Combine 9x and -4x to get 5x.

$$2x^{2}+5x-18=0$$

Add 18 to both sides. Anything plus zero gives itself.

$$2x^{2}+5x=18$$

Divide both sides by 2.

$$\frac{2x^{2}+5x}{2}=\frac{18}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{5}{2}x=\frac{18}{2}$$

Divide 18 by 2.

$$x^{2}+\frac{5}{2}x=9$$

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=9+\left(\frac{5}{4}\right)^{2}$$

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=9+\frac{25}{16}$$

Add 9 to \frac{25}{16}.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{169}{16}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{13}{4} x+\frac{5}{4}=-\frac{13}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

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