Solve -left(3x-5right)left(5x-1right)=0

$$- \left( 3x-5 \right) \left( 5x-1 \right) =0$$

Solve for x

$$x=\frac{1}{5}=0.2$$

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Steps Using Factoring By Grouping

To find the opposite of 3x-5, find the opposite of each term.

$$\left(-3x-\left(-5\right)\right)\left(5x-1\right)=0$$

The opposite of -5 is 5.

$$\left(-3x+5\right)\left(5x-1\right)=0$$

Apply the distributive property by multiplying each term of -3x+5 by each term of 5x-1.

$$-15x^{2}+3x+25x-5=0$$

Combine 3x and 25x to get 28x.

$$-15x^{2}+28x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -15x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=28 ab=-15\left(-5\right)=75$$

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 75.

$$1,75 3,25 5,15$$

Calculate the sum for each pair.

$$1+75=76 3+25=28 5+15=20$$

The solution is the pair that gives sum 28.

$$a=25 b=3$$

Rewrite -15x^{2}+28x-5 as \left(-15x^{2}+25x\right)+\left(3x-5\right).

$$\left(-15x^{2}+25x\right)+\left(3x-5\right)$$

Factor out -5x in -15x^{2}+25x.

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

Factor out common term 3x-5 by using distributive property.

$$\left(3x-5\right)\left(-5x+1\right)$$

To find equation solutions, solve 3x-5=0 and -5x+1=0.

$$x=\frac{5}{3} x=\frac{1}{5}$$

Steps Using the Quadratic Formula

To find the opposite of 3x-5, find the opposite of each term.

$$\left(-3x-\left(-5\right)\right)\left(5x-1\right)=0$$

The opposite of -5 is 5.

$$\left(-3x+5\right)\left(5x-1\right)=0$$

Apply the distributive property by multiplying each term of -3x+5 by each term of 5x-1.

$$-15x^{2}+3x+25x-5=0$$

Combine 3x and 25x to get 28x.

$$-15x^{2}+28x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 28 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-15\right)\left(-5\right)}}{2\left(-15\right)}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\left(-15\right)\left(-5\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-28±\sqrt{784+60\left(-5\right)}}{2\left(-15\right)}$$

Multiply 60 times -5.

$$x=\frac{-28±\sqrt{784-300}}{2\left(-15\right)}$$

Add 784 to -300.

$$x=\frac{-28±\sqrt{484}}{2\left(-15\right)}$$

Take the square root of 484.

$$x=\frac{-28±22}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-28±22}{-30}$$

Now solve the equation x=\frac{-28±22}{-30} when ± is plus. Add -28 to 22.

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

Reduce the fraction \frac{-6}{-30} to lowest terms by extracting and canceling out 6.

$$x=\frac{1}{5}$$

Now solve the equation x=\frac{-28±22}{-30} when ± is minus. Subtract 22 from -28.

$$x=-\frac{50}{-30}$$

Reduce the fraction \frac{-50}{-30} to lowest terms by extracting and canceling out 10.

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

The equation is now solved.

$$x=\frac{1}{5} x=\frac{5}{3}$$

Steps for Completing the Square

To find the opposite of 3x-5, find the opposite of each term.

$$\left(-3x-\left(-5\right)\right)\left(5x-1\right)=0$$

The opposite of -5 is 5.

$$\left(-3x+5\right)\left(5x-1\right)=0$$

Apply the distributive property by multiplying each term of -3x+5 by each term of 5x-1.

$$-15x^{2}+3x+25x-5=0$$

Combine 3x and 25x to get 28x.

$$-15x^{2}+28x-5=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$-15x^{2}+28x=5$$

Divide both sides by -15.

$$\frac{-15x^{2}+28x}{-15}=\frac{5}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{28}{-15}x=\frac{5}{-15}$$

Divide 28 by -15.

$$x^{2}-\frac{28}{15}x=\frac{5}{-15}$$

Reduce the fraction \frac{5}{-15} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{28}{15}x=-\frac{1}{3}$$

Divide -\frac{28}{15}, the coefficient of the x term, by 2 to get -\frac{14}{15}. Then add the square of -\frac{14}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{28}{15}x+\left(-\frac{14}{15}\right)^{2}=-\frac{1}{3}+\left(-\frac{14}{15}\right)^{2}$$

Square -\frac{14}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=-\frac{1}{3}+\frac{196}{225}$$

Add -\frac{1}{3} to \frac{196}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{121}{225}$$

Factor x^{2}-\frac{28}{15}x+\frac{196}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{14}{15}\right)^{2}=\frac{121}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{14}{15}\right)^{2}}=\sqrt{\frac{121}{225}}$$

Simplify.

$$x-\frac{14}{15}=\frac{11}{15} x-\frac{14}{15}=-\frac{11}{15}$$

Add \frac{14}{15} to both sides of the equation.

$$x=\frac{5}{3} x=\frac{1}{5}$$