Solve 9x^2-4x^2+18x+16x+29=0

$$9 x ^ { 2 } - 4 x ^ { 2 } + 18 x + 16 x + 29 = 0$$

Solve for x

$$x = -\frac{29}{5} = -5\frac{4}{5} = -5.8$$

$$x=-1$$

Steps Using Factoring By Grouping

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}+18x+16x+29=0$$

Combine 18x and 16x to get 34x.

$$5x^{2}+34x+29=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx+29. To find a and b, set up a system to be solved.

$$a+b=34 ab=5\times 29=145$$

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 145.

$$1,145 5,29$$

Calculate the sum for each pair.

$$1+145=146 5+29=34$$

The solution is the pair that gives sum 34.

$$a=5 b=29$$

Rewrite 5x^{2}+34x+29 as \left(5x^{2}+5x\right)+\left(29x+29\right).

$$\left(5x^{2}+5x\right)+\left(29x+29\right)$$

Factor out 5x in the first and 29 in the second group.

$$5x\left(x+1\right)+29\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(5x+29\right)$$

To find equation solutions, solve x+1=0 and 5x+29=0.

$$x=-1 x=-\frac{29}{5}$$

Steps Using the Quadratic Formula

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}+18x+16x+29=0$$

Combine 18x and 16x to get 34x.

$$5x^{2}+34x+29=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 34 for b, and 29 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\times 5\times 29}}{2\times 5}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\times 5\times 29}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-34±\sqrt{1156-20\times 29}}{2\times 5}$$

Multiply -20 times 29.

$$x=\frac{-34±\sqrt{1156-580}}{2\times 5}$$

Add 1156 to -580.

$$x=\frac{-34±\sqrt{576}}{2\times 5}$$

Take the square root of 576.

$$x=\frac{-34±24}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-34±24}{10}$$

Now solve the equation x=\frac{-34±24}{10} when ± is plus. Add -34 to 24.

$$x=-\frac{10}{10}$$

Divide -10 by 10.

$$x=-1$$

Now solve the equation x=\frac{-34±24}{10} when ± is minus. Subtract 24 from -34.

$$x=-\frac{58}{10}$$

Reduce the fraction \frac{-58}{10} to lowest terms by extracting and canceling out 2.

$$x=-\frac{29}{5}$$

The equation is now solved.

$$x=-1 x=-\frac{29}{5}$$

Steps for Completing the Square

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}+18x+16x+29=0$$

Combine 18x and 16x to get 34x.

$$5x^{2}+34x+29=0$$

Subtract 29 from both sides. Anything subtracted from zero gives its negation.

$$5x^{2}+34x=-29$$

Divide both sides by 5.

$$\frac{5x^{2}+34x}{5}=-\frac{29}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{34}{5}x=-\frac{29}{5}$$

Divide \frac{34}{5}, the coefficient of the x term, by 2 to get \frac{17}{5}. Then add the square of \frac{17}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{34}{5}x+\left(\frac{17}{5}\right)^{2}=-\frac{29}{5}+\left(\frac{17}{5}\right)^{2}$$

Square \frac{17}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{34}{5}x+\frac{289}{25}=-\frac{29}{5}+\frac{289}{25}$$

Add -\frac{29}{5} to \frac{289}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{34}{5}x+\frac{289}{25}=\frac{144}{25}$$

Factor x^{2}+\frac{34}{5}x+\frac{289}{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}.

$$\left(x+\frac{17}{5}\right)^{2}=\frac{144}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{5}\right)^{2}}=\sqrt{\frac{144}{25}}$$

Simplify.

$$x+\frac{17}{5}=\frac{12}{5} x+\frac{17}{5}=-\frac{12}{5}$$

Subtract \frac{17}{5} from both sides of the equation.

$$x=-1 x=-\frac{29}{5}$$