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

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

Solve for x

$$x=-\frac{1}{7}\approx -0.142857143$$

$$x=1$$

Steps Using Factoring By Grouping

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

$$2x^{2}+x-3+\left(x-1\right)\left(5x-2\right)=0$$

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

$$2x^{2}+x-3+5x^{2}-7x+2=0$$

Combine 2x^{2} and 5x^{2} to get 7x^{2}.

$$7x^{2}+x-3-7x+2=0$$

Combine x and -7x to get -6x.

$$7x^{2}-6x-3+2=0$$

Add -3 and 2 to get -1.

$$7x^{2}-6x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-6 ab=7\left(-1\right)=-7$$

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. The only such pair is the system solution.

$$a=-7 b=1$$

Rewrite 7x^{2}-6x-1 as \left(7x^{2}-7x\right)+\left(x-1\right).

$$\left(7x^{2}-7x\right)+\left(x-1\right)$$

Factor out 7x in 7x^{2}-7x.

$$7x\left(x-1\right)+x-1$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(7x+1\right)$$

To find equation solutions, solve x-1=0 and 7x+1=0.

$$x=1 x=-\frac{1}{7}$$

Steps Using the Quadratic Formula

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

$$2x^{2}+x-3+\left(x-1\right)\left(5x-2\right)=0$$

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

$$2x^{2}+x-3+5x^{2}-7x+2=0$$

Combine 2x^{2} and 5x^{2} to get 7x^{2}.

$$7x^{2}+x-3-7x+2=0$$

Combine x and -7x to get -6x.

$$7x^{2}-6x-3+2=0$$

Add -3 and 2 to get -1.

$$7x^{2}-6x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -6 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 7\left(-1\right)}}{2\times 7}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 7\left(-1\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-6\right)±\sqrt{36-28\left(-1\right)}}{2\times 7}$$

Multiply -28 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+28}}{2\times 7}$$

Add 36 to 28.

$$x=\frac{-\left(-6\right)±\sqrt{64}}{2\times 7}$$

Take the square root of 64.

$$x=\frac{-\left(-6\right)±8}{2\times 7}$$

The opposite of -6 is 6.

$$x=\frac{6±8}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{6±8}{14}$$

Now solve the equation x=\frac{6±8}{14} when ± is plus. Add 6 to 8.

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

Divide 14 by 14.

$$x=1$$

Now solve the equation x=\frac{6±8}{14} when ± is minus. Subtract 8 from 6.

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

Reduce the fraction \frac{-2}{14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{7}$$

The equation is now solved.

$$x=1 x=-\frac{1}{7}$$

Steps for Completing the Square

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

$$2x^{2}+x-3+\left(x-1\right)\left(5x-2\right)=0$$

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

$$2x^{2}+x-3+5x^{2}-7x+2=0$$

Combine 2x^{2} and 5x^{2} to get 7x^{2}.

$$7x^{2}+x-3-7x+2=0$$

Combine x and -7x to get -6x.

$$7x^{2}-6x-3+2=0$$

Add -3 and 2 to get -1.

$$7x^{2}-6x-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$7x^{2}-6x=1$$

Divide both sides by 7.

$$\frac{7x^{2}-6x}{7}=\frac{1}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{6}{7}x=\frac{1}{7}$$

Divide -\frac{6}{7}, the coefficient of the x term, by 2 to get -\frac{3}{7}. Then add the square of -\frac{3}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6}{7}x+\left(-\frac{3}{7}\right)^{2}=\frac{1}{7}+\left(-\frac{3}{7}\right)^{2}$$

Square -\frac{3}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6}{7}x+\frac{9}{49}=\frac{1}{7}+\frac{9}{49}$$

Add \frac{1}{7} to \frac{9}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6}{7}x+\frac{9}{49}=\frac{16}{49}$$

Factor x^{2}-\frac{6}{7}x+\frac{9}{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}.

$$\left(x-\frac{3}{7}\right)^{2}=\frac{16}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{7}\right)^{2}}=\sqrt{\frac{16}{49}}$$

Simplify.

$$x-\frac{3}{7}=\frac{4}{7} x-\frac{3}{7}=-\frac{4}{7}$$

Add \frac{3}{7} to both sides of the equation.

$$x=1 x=-\frac{1}{7}$$