Solve left(5xright)^2+30x+8=0

$${ \left(5x \right) }^{ 2 } +30x+8=0$$

Solve for x

$$x=-\frac{4}{5}=-0.8$$

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

Steps Using Factoring By Grouping

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}+30x+8=0$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+30x+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 25x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=30 ab=25\times 8=200$$

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 200.

$$1,200 2,100 4,50 5,40 8,25 10,20$$

Calculate the sum for each pair.

$$1+200=201 2+100=102 4+50=54 5+40=45 8+25=33 10+20=30$$

The solution is the pair that gives sum 30.

$$a=10 b=20$$

Rewrite 25x^{2}+30x+8 as \left(25x^{2}+10x\right)+\left(20x+8\right).

$$\left(25x^{2}+10x\right)+\left(20x+8\right)$$

Factor out 5x in the first and 4 in the second group.

$$5x\left(5x+2\right)+4\left(5x+2\right)$$

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

$$\left(5x+2\right)\left(5x+4\right)$$

To find equation solutions, solve 5x+2=0 and 5x+4=0.

$$x=-\frac{2}{5} x=-\frac{4}{5}$$

Steps Using the Quadratic Formula

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}+30x+8=0$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+30x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, 30 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 25\times 8}}{2\times 25}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 25\times 8}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-30±\sqrt{900-100\times 8}}{2\times 25}$$

Multiply -100 times 8.

$$x=\frac{-30±\sqrt{900-800}}{2\times 25}$$

Add 900 to -800.

$$x=\frac{-30±\sqrt{100}}{2\times 25}$$

Take the square root of 100.

$$x=\frac{-30±10}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-30±10}{50}$$

Now solve the equation x=\frac{-30±10}{50} when ± is plus. Add -30 to 10.

$$x=-\frac{20}{50}$$

Reduce the fraction \frac{-20}{50} to lowest terms by extracting and canceling out 10.

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

Now solve the equation x=\frac{-30±10}{50} when ± is minus. Subtract 10 from -30.

$$x=-\frac{40}{50}$$

Reduce the fraction \frac{-40}{50} to lowest terms by extracting and canceling out 10.

$$x=-\frac{4}{5}$$

The equation is now solved.

$$x=-\frac{2}{5} x=-\frac{4}{5}$$

Steps for Completing the Square

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}+30x+8=0$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+30x+8=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$25x^{2}+30x=-8$$

Divide both sides by 25.

$$\frac{25x^{2}+30x}{25}=-\frac{8}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\frac{30}{25}x=-\frac{8}{25}$$

Reduce the fraction \frac{30}{25} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{6}{5}x=-\frac{8}{25}$$

Divide \frac{6}{5}, the coefficient of the x term, by 2 to get \frac{3}{5}. Then add the square of \frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{6}{5}x+\left(\frac{3}{5}\right)^{2}=-\frac{8}{25}+\left(\frac{3}{5}\right)^{2}$$

Square \frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{-8+9}{25}$$

Add -\frac{8}{25} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{6}{5}x+\frac{9}{25}=\frac{1}{25}$$

Factor x^{2}+\frac{6}{5}x+\frac{9}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{5}\right)^{2}=\frac{1}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{5}\right)^{2}}=\sqrt{\frac{1}{25}}$$

Simplify.

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

Subtract \frac{3}{5} from both sides of the equation.

$$x=-\frac{2}{5} x=-\frac{4}{5}$$