Solve 3x-15=2x(x-5)

$$3 x - 15 = 2 x ( x - 5 )$$

Solve for x

$$x=5$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 2x by x-5.

$$3x-15=2x^{2}-10x$$

Subtract 2x^{2} from both sides.

$$3x-15-2x^{2}=-10x$$

Add 10x to both sides.

$$3x-15-2x^{2}+10x=0$$

Combine 3x and 10x to get 13x.

$$13x-15-2x^{2}=0$$

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

$$-2x^{2}+13x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=13 ab=-2\left(-15\right)=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 13.

$$a=10 b=3$$

Rewrite -2x^{2}+13x-15 as \left(-2x^{2}+10x\right)+\left(3x-15\right).

$$\left(-2x^{2}+10x\right)+\left(3x-15\right)$$

Factor out 2x in the first and -3 in the second group.

$$2x\left(-x+5\right)-3\left(-x+5\right)$$

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

$$\left(-x+5\right)\left(2x-3\right)$$

To find equation solutions, solve -x+5=0 and 2x-3=0.

$$x=5 x=\frac{3}{2}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by x-5.

$$3x-15=2x^{2}-10x$$

Subtract 2x^{2} from both sides.

$$3x-15-2x^{2}=-10x$$

Add 10x to both sides.

$$3x-15-2x^{2}+10x=0$$

Combine 3x and 10x to get 13x.

$$13x-15-2x^{2}=0$$

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.

$$-2x^{2}+13x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 13 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-13±\sqrt{169+8\left(-15\right)}}{2\left(-2\right)}$$

Multiply 8 times -15.

$$x=\frac{-13±\sqrt{169-120}}{2\left(-2\right)}$$

Add 169 to -120.

$$x=\frac{-13±\sqrt{49}}{2\left(-2\right)}$$

Take the square root of 49.

$$x=\frac{-13±7}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-13±7}{-4}$$

Now solve the equation x=\frac{-13±7}{-4} when ± is plus. Add -13 to 7.

$$x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-13±7}{-4} when ± is minus. Subtract 7 from -13.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

The equation is now solved.

$$x=\frac{3}{2} x=5$$

Steps for Completing the Square

Use the distributive property to multiply 2x by x-5.

$$3x-15=2x^{2}-10x$$

Subtract 2x^{2} from both sides.

$$3x-15-2x^{2}=-10x$$

Add 10x to both sides.

$$3x-15-2x^{2}+10x=0$$

Combine 3x and 10x to get 13x.

$$13x-15-2x^{2}=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$13x-2x^{2}=15$$

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.

$$-2x^{2}+13x=15$$

Divide both sides by -2.

$$\frac{-2x^{2}+13x}{-2}=\frac{15}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{13}{-2}x=\frac{15}{-2}$$

Divide 13 by -2.

$$x^{2}-\frac{13}{2}x=\frac{15}{-2}$$

Divide 15 by -2.

$$x^{2}-\frac{13}{2}x=-\frac{15}{2}$$

Divide -\frac{13}{2}, the coefficient of the x term, by 2 to get -\frac{13}{4}. Then add the square of -\frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-\frac{15}{2}+\left(-\frac{13}{4}\right)^{2}$$

Square -\frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{15}{2}+\frac{169}{16}$$

Add -\frac{15}{2} to \frac{169}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{49}{16}$$

Factor x^{2}-\frac{13}{2}x+\frac{169}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{13}{4}=\frac{7}{4} x-\frac{13}{4}=-\frac{7}{4}$$

Add \frac{13}{4} to both sides of the equation.

$$x=5 x=\frac{3}{2}$$