Solve 4(1+9x^2+6x)=56x+84

$$4(1+9 { x }^{ 2 } +6x)=56x+84$$

Solve for x

$$x = -\frac{10}{9} = -1\frac{1}{9} \approx -1.111111111$$

$$x=2$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 4 by 1+9x^{2}+6x.

$$4+36x^{2}+24x=56x+84$$

Subtract 56x from both sides.

$$4+36x^{2}+24x-56x=84$$

Combine 24x and -56x to get -32x.

$$4+36x^{2}-32x=84$$

Subtract 84 from both sides.

$$4+36x^{2}-32x-84=0$$

Subtract 84 from 4 to get -80.

$$-80+36x^{2}-32x=0$$

Divide both sides by 4.

$$-20+9x^{2}-8x=0$$

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

$$9x^{2}-8x-20=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx-20. To find a and b, set up a system to be solved.

$$a+b=-8 ab=9\left(-20\right)=-180$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Calculate the sum for each pair.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

The solution is the pair that gives sum -8.

$$a=-18 b=10$$

Rewrite 9x^{2}-8x-20 as \left(9x^{2}-18x\right)+\left(10x-20\right).

$$\left(9x^{2}-18x\right)+\left(10x-20\right)$$

Factor out 9x in the first and 10 in the second group.

$$9x\left(x-2\right)+10\left(x-2\right)$$

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

$$\left(x-2\right)\left(9x+10\right)$$

To find equation solutions, solve x-2=0 and 9x+10=0.

$$x=2 x=-\frac{10}{9}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by 1+9x^{2}+6x.

$$4+36x^{2}+24x=56x+84$$

Subtract 56x from both sides.

$$4+36x^{2}+24x-56x=84$$

Combine 24x and -56x to get -32x.

$$4+36x^{2}-32x=84$$

Subtract 84 from both sides.

$$4+36x^{2}-32x-84=0$$

Subtract 84 from 4 to get -80.

$$-80+36x^{2}-32x=0$$

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.

$$36x^{2}-32x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 36 for a, -32 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 36\left(-80\right)}}{2\times 36}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 36\left(-80\right)}}{2\times 36}$$

Multiply -4 times 36.

$$x=\frac{-\left(-32\right)±\sqrt{1024-144\left(-80\right)}}{2\times 36}$$

Multiply -144 times -80.

$$x=\frac{-\left(-32\right)±\sqrt{1024+11520}}{2\times 36}$$

Add 1024 to 11520.

$$x=\frac{-\left(-32\right)±\sqrt{12544}}{2\times 36}$$

Take the square root of 12544.

$$x=\frac{-\left(-32\right)±112}{2\times 36}$$

The opposite of -32 is 32.

$$x=\frac{32±112}{2\times 36}$$

Multiply 2 times 36.

$$x=\frac{32±112}{72}$$

Now solve the equation x=\frac{32±112}{72} when ± is plus. Add 32 to 112.

$$x=\frac{144}{72}$$

Divide 144 by 72.

$$x=2$$

Now solve the equation x=\frac{32±112}{72} when ± is minus. Subtract 112 from 32.

$$x=-\frac{80}{72}$$

Reduce the fraction \frac{-80}{72} to lowest terms by extracting and canceling out 8.

$$x=-\frac{10}{9}$$

The equation is now solved.

$$x=2 x=-\frac{10}{9}$$

Steps for Completing the Square

Use the distributive property to multiply 4 by 1+9x^{2}+6x.

$$4+36x^{2}+24x=56x+84$$

Subtract 56x from both sides.

$$4+36x^{2}+24x-56x=84$$

Combine 24x and -56x to get -32x.

$$4+36x^{2}-32x=84$$

Subtract 4 from both sides.

$$36x^{2}-32x=84-4$$

Subtract 4 from 84 to get 80.

$$36x^{2}-32x=80$$

Divide both sides by 36.

$$\frac{36x^{2}-32x}{36}=\frac{80}{36}$$

Dividing by 36 undoes the multiplication by 36.

$$x^{2}+\left(-\frac{32}{36}\right)x=\frac{80}{36}$$

Reduce the fraction \frac{-32}{36} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{8}{9}x=\frac{80}{36}$$

Reduce the fraction \frac{80}{36} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{8}{9}x=\frac{20}{9}$$

Divide -\frac{8}{9}, the coefficient of the x term, by 2 to get -\frac{4}{9}. Then add the square of -\frac{4}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{9}x+\left(-\frac{4}{9}\right)^{2}=\frac{20}{9}+\left(-\frac{4}{9}\right)^{2}$$

Square -\frac{4}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{20}{9}+\frac{16}{81}$$

Add \frac{20}{9} to \frac{16}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{9}x+\frac{16}{81}=\frac{196}{81}$$

Factor x^{2}-\frac{8}{9}x+\frac{16}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{9}\right)^{2}=\frac{196}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{9}\right)^{2}}=\sqrt{\frac{196}{81}}$$

Simplify.

$$x-\frac{4}{9}=\frac{14}{9} x-\frac{4}{9}=-\frac{14}{9}$$

Add \frac{4}{9} to both sides of the equation.

$$x=2 x=-\frac{10}{9}$$