Solve 35-36x=-9x^2

$$35 - 36 x = - 9 x ^ { 2 }$$

Solve for x

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

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps Using Factoring By Grouping

Add 9x^{2} to both sides.

$$35-36x+9x^{2}=0$$

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

$$9x^{2}-36x+35=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx+35. To find a and b, set up a system to be solved.

$$a+b=-36 ab=9\times 35=315$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 315.

$$-1,-315 -3,-105 -5,-63 -7,-45 -9,-35 -15,-21$$

Calculate the sum for each pair.

$$-1-315=-316 -3-105=-108 -5-63=-68 -7-45=-52 -9-35=-44 -15-21=-36$$

The solution is the pair that gives sum -36.

$$a=-21 b=-15$$

Rewrite 9x^{2}-36x+35 as \left(9x^{2}-21x\right)+\left(-15x+35\right).

$$\left(9x^{2}-21x\right)+\left(-15x+35\right)$$

Factor out 3x in the first and -5 in the second group.

$$3x\left(3x-7\right)-5\left(3x-7\right)$$

Factor out common term 3x-7 by using distributive property.

$$\left(3x-7\right)\left(3x-5\right)$$

To find equation solutions, solve 3x-7=0 and 3x-5=0.

$$x=\frac{7}{3} x=\frac{5}{3}$$

Steps Using the Quadratic Formula

Add 9x^{2} to both sides.

$$35-36x+9x^{2}=0$$

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.

$$9x^{2}-36x+35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -36 for b, and 35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 9\times 35}}{2\times 9}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 9\times 35}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-36\right)±\sqrt{1296-36\times 35}}{2\times 9}$$

Multiply -36 times 35.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1260}}{2\times 9}$$

Add 1296 to -1260.

$$x=\frac{-\left(-36\right)±\sqrt{36}}{2\times 9}$$

Take the square root of 36.

$$x=\frac{-\left(-36\right)±6}{2\times 9}$$

The opposite of -36 is 36.

$$x=\frac{36±6}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{36±6}{18}$$

Now solve the equation x=\frac{36±6}{18} when ± is plus. Add 36 to 6.

$$x=\frac{42}{18}$$

Reduce the fraction \frac{42}{18} to lowest terms by extracting and canceling out 6.

$$x=\frac{7}{3}$$

Now solve the equation x=\frac{36±6}{18} when ± is minus. Subtract 6 from 36.

$$x=\frac{30}{18}$$

Reduce the fraction \frac{30}{18} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

$$x=\frac{7}{3} x=\frac{5}{3}$$

Steps for Completing the Square

Add 9x^{2} to both sides.

$$35-36x+9x^{2}=0$$

Subtract 35 from both sides. Anything subtracted from zero gives its negation.

$$-36x+9x^{2}=-35$$

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.

$$9x^{2}-36x=-35$$

Divide both sides by 9.

$$\frac{9x^{2}-36x}{9}=-\frac{35}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{36}{9}\right)x=-\frac{35}{9}$$

Divide -36 by 9.

$$x^{2}-4x=-\frac{35}{9}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{35}{9}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-\frac{35}{9}+4$$

Add -\frac{35}{9} to 4.

$$x^{2}-4x+4=\frac{1}{9}$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{1}{9}}$$

Simplify.

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

Add 2 to both sides of the equation.

$$x=\frac{7}{3} x=\frac{5}{3}$$