Solve 0=left(5x-2right)left(7x+5right)

$$0= \left( 5x-2 \right) \left( 7x+5 \right)$$

Solve for x

$$x=-\frac{5}{7}\approx -0.714285714$$

$$x=\frac{2}{5}=0.4$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 5x-2 by 7x+5 and combine like terms.

$$0=35x^{2}+11x-10$$

Swap sides so that all variable terms are on the left hand side.

$$35x^{2}+11x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 35x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=11 ab=35\left(-10\right)=-350$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -350.

$$-1,350 -2,175 -5,70 -7,50 -10,35 -14,25$$

Calculate the sum for each pair.

$$-1+350=349 -2+175=173 -5+70=65 -7+50=43 -10+35=25 -14+25=11$$

The solution is the pair that gives sum 11.

$$a=-14 b=25$$

Rewrite 35x^{2}+11x-10 as \left(35x^{2}-14x\right)+\left(25x-10\right).

$$\left(35x^{2}-14x\right)+\left(25x-10\right)$$

Factor out 7x in the first and 5 in the second group.

$$7x\left(5x-2\right)+5\left(5x-2\right)$$

Factor out common term 5x-2 by using distributive property.

$$\left(5x-2\right)\left(7x+5\right)$$

To find equation solutions, solve 5x-2=0 and 7x+5=0.

$$x=\frac{2}{5} x=-\frac{5}{7}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x-2 by 7x+5 and combine like terms.

$$0=35x^{2}+11x-10$$

Swap sides so that all variable terms are on the left hand side.

$$35x^{2}+11x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 35 for a, 11 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 35\left(-10\right)}}{2\times 35}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 35\left(-10\right)}}{2\times 35}$$

Multiply -4 times 35.

$$x=\frac{-11±\sqrt{121-140\left(-10\right)}}{2\times 35}$$

Multiply -140 times -10.

$$x=\frac{-11±\sqrt{121+1400}}{2\times 35}$$

Add 121 to 1400.

$$x=\frac{-11±\sqrt{1521}}{2\times 35}$$

Take the square root of 1521.

$$x=\frac{-11±39}{2\times 35}$$

Multiply 2 times 35.

$$x=\frac{-11±39}{70}$$

Now solve the equation x=\frac{-11±39}{70} when ± is plus. Add -11 to 39.

$$x=\frac{28}{70}$$

Reduce the fraction \frac{28}{70} to lowest terms by extracting and canceling out 14.

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

Now solve the equation x=\frac{-11±39}{70} when ± is minus. Subtract 39 from -11.

$$x=-\frac{50}{70}$$

Reduce the fraction \frac{-50}{70} to lowest terms by extracting and canceling out 10.

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

The equation is now solved.

$$x=\frac{2}{5} x=-\frac{5}{7}$$

Steps for Completing the Square

Use the distributive property to multiply 5x-2 by 7x+5 and combine like terms.

$$0=35x^{2}+11x-10$$

Swap sides so that all variable terms are on the left hand side.

$$35x^{2}+11x-10=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$35x^{2}+11x=10$$

Divide both sides by 35.

$$\frac{35x^{2}+11x}{35}=\frac{10}{35}$$

Dividing by 35 undoes the multiplication by 35.

$$x^{2}+\frac{11}{35}x=\frac{10}{35}$$

Reduce the fraction \frac{10}{35} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{11}{35}x=\frac{2}{7}$$

Divide \frac{11}{35}, the coefficient of the x term, by 2 to get \frac{11}{70}. Then add the square of \frac{11}{70} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{35}x+\left(\frac{11}{70}\right)^{2}=\frac{2}{7}+\left(\frac{11}{70}\right)^{2}$$

Square \frac{11}{70} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{35}x+\frac{121}{4900}=\frac{2}{7}+\frac{121}{4900}$$

Add \frac{2}{7} to \frac{121}{4900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{35}x+\frac{121}{4900}=\frac{1521}{4900}$$

Factor x^{2}+\frac{11}{35}x+\frac{121}{4900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{70}\right)^{2}=\frac{1521}{4900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{70}\right)^{2}}=\sqrt{\frac{1521}{4900}}$$

Simplify.

$$x+\frac{11}{70}=\frac{39}{70} x+\frac{11}{70}=-\frac{39}{70}$$

Subtract \frac{11}{70} from both sides of the equation.

$$x=\frac{2}{5} x=-\frac{5}{7}$$