Solve 6x-10+x-1=3(x-1)(3x-5)

$$6 x - 10 + x - 1 = 3 ( x - 1 ) ( 3 x - 5 )$$

Solve for x

$$x=2$$

$$x = \frac{13}{9} = 1\frac{4}{9} \approx 1.444444444$$

Steps Using the Quadratic Formula

Combine 6x and x to get 7x.

$$7x-10-1=3\left(x-1\right)\left(3x-5\right)$$

Subtract 1 from -10 to get -11.

$$7x-11=3\left(x-1\right)\left(3x-5\right)$$

Use the distributive property to multiply 3 by x-1.

$$7x-11=\left(3x-3\right)\left(3x-5\right)$$

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

$$7x-11=9x^{2}-24x+15$$

Subtract 9x^{2} from both sides.

$$7x-11-9x^{2}=-24x+15$$

Add 24x to both sides.

$$7x-11-9x^{2}+24x=15$$

Combine 7x and 24x to get 31x.

$$31x-11-9x^{2}=15$$

Subtract 15 from both sides.

$$31x-11-9x^{2}-15=0$$

Subtract 15 from -11 to get -26.

$$31x-26-9x^{2}=0$$

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.

$$-9x^{2}+31x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 31 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\left(-9\right)\left(-26\right)}}{2\left(-9\right)}$$

Square 31.

$$x=\frac{-31±\sqrt{961-4\left(-9\right)\left(-26\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-31±\sqrt{961+36\left(-26\right)}}{2\left(-9\right)}$$

Multiply 36 times -26.

$$x=\frac{-31±\sqrt{961-936}}{2\left(-9\right)}$$

Add 961 to -936.

$$x=\frac{-31±\sqrt{25}}{2\left(-9\right)}$$

Take the square root of 25.

$$x=\frac{-31±5}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-31±5}{-18}$$

Now solve the equation x=\frac{-31±5}{-18} when ± is plus. Add -31 to 5.

$$x=-\frac{26}{-18}$$

Reduce the fraction \frac{-26}{-18} to lowest terms by extracting and canceling out 2.

$$x=\frac{13}{9}$$

Now solve the equation x=\frac{-31±5}{-18} when ± is minus. Subtract 5 from -31.

$$x=-\frac{36}{-18}$$

Divide -36 by -18.

$$x=2$$

The equation is now solved.

$$x=\frac{13}{9} x=2$$

Steps for Completing the Square

Combine 6x and x to get 7x.

$$7x-10-1=3\left(x-1\right)\left(3x-5\right)$$

Subtract 1 from -10 to get -11.

$$7x-11=3\left(x-1\right)\left(3x-5\right)$$

Use the distributive property to multiply 3 by x-1.

$$7x-11=\left(3x-3\right)\left(3x-5\right)$$

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

$$7x-11=9x^{2}-24x+15$$

Subtract 9x^{2} from both sides.

$$7x-11-9x^{2}=-24x+15$$

Add 24x to both sides.

$$7x-11-9x^{2}+24x=15$$

Combine 7x and 24x to get 31x.

$$31x-11-9x^{2}=15$$

Add 11 to both sides.

$$31x-9x^{2}=15+11$$

Add 15 and 11 to get 26.

$$31x-9x^{2}=26$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-9x^{2}+31x=26$$

Divide both sides by -9.

$$\frac{-9x^{2}+31x}{-9}=\frac{26}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{31}{-9}x=\frac{26}{-9}$$

Divide 31 by -9.

$$x^{2}-\frac{31}{9}x=\frac{26}{-9}$$

Divide 26 by -9.

$$x^{2}-\frac{31}{9}x=-\frac{26}{9}$$

Divide -\frac{31}{9}, the coefficient of the x term, by 2 to get -\frac{31}{18}. Then add the square of -\frac{31}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{31}{9}x+\left(-\frac{31}{18}\right)^{2}=-\frac{26}{9}+\left(-\frac{31}{18}\right)^{2}$$

Square -\frac{31}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{31}{9}x+\frac{961}{324}=-\frac{26}{9}+\frac{961}{324}$$

Add -\frac{26}{9} to \frac{961}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{31}{9}x+\frac{961}{324}=\frac{25}{324}$$

Factor x^{2}-\frac{31}{9}x+\frac{961}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{18}\right)^{2}=\frac{25}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{18}\right)^{2}}=\sqrt{\frac{25}{324}}$$

Simplify.

$$x-\frac{31}{18}=\frac{5}{18} x-\frac{31}{18}=-\frac{5}{18}$$

Add \frac{31}{18} to both sides of the equation.

$$x=2 x=\frac{13}{9}$$