Solve -7(x-5)+3x(x-5)=0

$$- 7 ( x - 5 ) + 3 x ( x - 5 ) = 0$$

Solve for x

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

$$x=5$$

Steps Using Factoring By Grouping

Use the distributive property to multiply -7 by x-5.

$$-7x+35+3x\left(x-5\right)=0$$

Use the distributive property to multiply 3x by x-5.

$$-7x+35+3x^{2}-15x=0$$

Combine -7x and -15x to get -22x.

$$-22x+35+3x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$3x^{2}-22x+35=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx+35. To find a and b, set up a system to be solved.

$$a+b=-22 ab=3\times 35=105$$

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 105.

$$-1,-105 -3,-35 -5,-21 -7,-15$$

Calculate the sum for each pair.

$$-1-105=-106 -3-35=-38 -5-21=-26 -7-15=-22$$

The solution is the pair that gives sum -22.

$$a=-15 b=-7$$

Rewrite 3x^{2}-22x+35 as \left(3x^{2}-15x\right)+\left(-7x+35\right).

$$\left(3x^{2}-15x\right)+\left(-7x+35\right)$$

Factor out 3x in the first and -7 in the second group.

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

Factor out common term x-5 by using distributive property.

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

To find equation solutions, solve x-5=0 and 3x-7=0.

$$x=5 x=\frac{7}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -7 by x-5.

$$-7x+35+3x\left(x-5\right)=0$$

Use the distributive property to multiply 3x by x-5.

$$-7x+35+3x^{2}-15x=0$$

Combine -7x and -15x to get -22x.

$$-22x+35+3x^{2}=0$$

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.

$$3x^{2}-22x+35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -22 for b, and 35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 3\times 35}}{2\times 3}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 3\times 35}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-22\right)±\sqrt{484-12\times 35}}{2\times 3}$$

Multiply -12 times 35.

$$x=\frac{-\left(-22\right)±\sqrt{484-420}}{2\times 3}$$

Add 484 to -420.

$$x=\frac{-\left(-22\right)±\sqrt{64}}{2\times 3}$$

Take the square root of 64.

$$x=\frac{-\left(-22\right)±8}{2\times 3}$$

The opposite of -22 is 22.

$$x=\frac{22±8}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{22±8}{6}$$

Now solve the equation x=\frac{22±8}{6} when ± is plus. Add 22 to 8.

$$x=\frac{30}{6}$$

Divide 30 by 6.

$$x=5$$

Now solve the equation x=\frac{22±8}{6} when ± is minus. Subtract 8 from 22.

$$x=\frac{14}{6}$$

Reduce the fraction \frac{14}{6} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=5 x=\frac{7}{3}$$

Steps for Completing the Square

Use the distributive property to multiply -7 by x-5.

$$-7x+35+3x\left(x-5\right)=0$$

Use the distributive property to multiply 3x by x-5.

$$-7x+35+3x^{2}-15x=0$$

Combine -7x and -15x to get -22x.

$$-22x+35+3x^{2}=0$$

Subtract 35 from both sides. Anything subtracted from zero gives its negation.

$$-22x+3x^{2}=-35$$

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.

$$3x^{2}-22x=-35$$

Divide both sides by 3.

$$\frac{3x^{2}-22x}{3}=-\frac{35}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{22}{3}x=-\frac{35}{3}$$

Divide -\frac{22}{3}, the coefficient of the x term, by 2 to get -\frac{11}{3}. Then add the square of -\frac{11}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{22}{3}x+\left(-\frac{11}{3}\right)^{2}=-\frac{35}{3}+\left(-\frac{11}{3}\right)^{2}$$

Square -\frac{11}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-\frac{35}{3}+\frac{121}{9}$$

Add -\frac{35}{3} to \frac{121}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=\frac{16}{9}$$

Factor x^{2}-\frac{22}{3}x+\frac{121}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{3}\right)^{2}=\frac{16}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{3}\right)^{2}}=\sqrt{\frac{16}{9}}$$

Simplify.

$$x-\frac{11}{3}=\frac{4}{3} x-\frac{11}{3}=-\frac{4}{3}$$

Add \frac{11}{3} to both sides of the equation.

$$x=5 x=\frac{7}{3}$$