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

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

Solve for x

$$x=\frac{1}{4}=0.25$$

$$x=2$$

Steps Using Factoring By Grouping

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

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

Use the distributive property to multiply x-2 by x-6 and combine like terms.

$$3x^{2}-x-10+x^{2}-8x+12=0$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-x-10-8x+12=0$$

Combine -x and -8x to get -9x.

$$4x^{2}-9x-10+12=0$$

Add -10 and 12 to get 2.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=-9 ab=4\times 2=8$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 8.

$$-1,-8 -2,-4$$

Calculate the sum for each pair.

$$-1-8=-9 -2-4=-6$$

The solution is the pair that gives sum -9.

$$a=-8 b=-1$$

Rewrite 4x^{2}-9x+2 as \left(4x^{2}-8x\right)+\left(-x+2\right).

$$\left(4x^{2}-8x\right)+\left(-x+2\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(x-2\right)-\left(x-2\right)$$

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

$$\left(x-2\right)\left(4x-1\right)$$

To find equation solutions, solve x-2=0 and 4x-1=0.

$$x=2 x=\frac{1}{4}$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply x-2 by x-6 and combine like terms.

$$3x^{2}-x-10+x^{2}-8x+12=0$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-x-10-8x+12=0$$

Combine -x and -8x to get -9x.

$$4x^{2}-9x-10+12=0$$

Add -10 and 12 to get 2.

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

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

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

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\times 2}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-9\right)±\sqrt{81-16\times 2}}{2\times 4}$$

Multiply -16 times 2.

$$x=\frac{-\left(-9\right)±\sqrt{81-32}}{2\times 4}$$

Add 81 to -32.

$$x=\frac{-\left(-9\right)±\sqrt{49}}{2\times 4}$$

Take the square root of 49.

$$x=\frac{-\left(-9\right)±7}{2\times 4}$$

The opposite of -9 is 9.

$$x=\frac{9±7}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{9±7}{8}$$

Now solve the equation x=\frac{9±7}{8} when ± is plus. Add 9 to 7.

$$x=\frac{16}{8}$$

Divide 16 by 8.

$$x=2$$

Now solve the equation x=\frac{9±7}{8} when ± is minus. Subtract 7 from 9.

$$x=\frac{2}{8}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{4}$$

The equation is now solved.

$$x=2 x=\frac{1}{4}$$

Steps for Completing the Square

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

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

Use the distributive property to multiply x-2 by x-6 and combine like terms.

$$3x^{2}-x-10+x^{2}-8x+12=0$$

Combine 3x^{2} and x^{2} to get 4x^{2}.

$$4x^{2}-x-10-8x+12=0$$

Combine -x and -8x to get -9x.

$$4x^{2}-9x-10+12=0$$

Add -10 and 12 to get 2.

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

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-9x=-2$$

Divide both sides by 4.

$$\frac{4x^{2}-9x}{4}=-\frac{2}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{9}{4}x=-\frac{2}{4}$$

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

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

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=-\frac{1}{2}+\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-\frac{1}{2}+\frac{81}{64}$$

Add -\frac{1}{2} to \frac{81}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{49}{64}$$

Factor x^{2}-\frac{9}{4}x+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{8}\right)^{2}=\frac{49}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Simplify.

$$x-\frac{9}{8}=\frac{7}{8} x-\frac{9}{8}=-\frac{7}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$x=2 x=\frac{1}{4}$$