Solve (2x-3)(5x+1)-(x-6)(x+6)+13x=0

$$( 2 x - 3 ) ( 5 x + 1 ) - ( x - 6 ) ( x + 6 ) + 13 x = 0$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{33}i}{3}\approx -0-1.914854216i$$

$$x=\frac{\sqrt{33}i}{3}\approx 1.914854216i$$

Steps by Finding Square Root

Use the distributive property to multiply 2x-3 by 5x+1 and combine like terms.

$$10x^{2}-13x-3-\left(x-6\right)\left(x+6\right)+13x=0$$

Consider \left(x-6\right)\left(x+6\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 6.

$$10x^{2}-13x-3-\left(x^{2}-36\right)+13x=0$$

To find the opposite of x^{2}-36, find the opposite of each term.

$$10x^{2}-13x-3-x^{2}+36+13x=0$$

Combine 10x^{2} and -x^{2} to get 9x^{2}.

$$9x^{2}-13x-3+36+13x=0$$

Add -3 and 36 to get 33.

$$9x^{2}-13x+33+13x=0$$

Combine -13x and 13x to get 0.

$$9x^{2}+33=0$$

Subtract 33 from both sides. Anything subtracted from zero gives its negation.

$$9x^{2}=-33$$

Divide both sides by 9.

$$x^{2}=\frac{-33}{9}$$

Reduce the fraction \frac{-33}{9} to lowest terms by extracting and canceling out 3.

$$x^{2}=-\frac{11}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{33}i}{3} x=-\frac{\sqrt{33}i}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-3 by 5x+1 and combine like terms.

$$10x^{2}-13x-3-\left(x-6\right)\left(x+6\right)+13x=0$$

Consider \left(x-6\right)\left(x+6\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 6.

$$10x^{2}-13x-3-\left(x^{2}-36\right)+13x=0$$

To find the opposite of x^{2}-36, find the opposite of each term.

$$10x^{2}-13x-3-x^{2}+36+13x=0$$

Combine 10x^{2} and -x^{2} to get 9x^{2}.

$$9x^{2}-13x-3+36+13x=0$$

Add -3 and 36 to get 33.

$$9x^{2}-13x+33+13x=0$$

Combine -13x and 13x to get 0.

$$9x^{2}+33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 0 for b, and 33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 9\times 33}}{2\times 9}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 9\times 33}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{0±\sqrt{-36\times 33}}{2\times 9}$$

Multiply -36 times 33.

$$x=\frac{0±\sqrt{-1188}}{2\times 9}$$

Take the square root of -1188.

$$x=\frac{0±6\sqrt{33}i}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{0±6\sqrt{33}i}{18}$$

Now solve the equation x=\frac{0±6\sqrt{33}i}{18} when ± is plus.

$$x=\frac{\sqrt{33}i}{3}$$

Now solve the equation x=\frac{0±6\sqrt{33}i}{18} when ± is minus.

$$x=-\frac{\sqrt{33}i}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{33}i}{3} x=-\frac{\sqrt{33}i}{3}$$