Solve 1*(9-6x+5x^2)*29=36-144x+144x^2

$$1 \cdot ( 9 - 6 x + 5 x ^ { 2 } ) \cdot 29 = 36 - 144 x + 144 x ^ { 2 }$$

Solve for x

$$x=15$$

Steps Using Factoring

Multiply 1 and 29 to get 29.

$$29\left(9-6x+5x^{2}\right)=36-144x+144x^{2}$$

Use the distributive property to multiply 29 by 9-6x+5x^{2}.

$$261-174x+145x^{2}=36-144x+144x^{2}$$

Subtract 36 from both sides.

$$261-174x+145x^{2}-36=-144x+144x^{2}$$

Subtract 36 from 261 to get 225.

$$225-174x+145x^{2}=-144x+144x^{2}$$

Add 144x to both sides.

$$225-174x+145x^{2}+144x=144x^{2}$$

Combine -174x and 144x to get -30x.

$$225-30x+145x^{2}=144x^{2}$$

Subtract 144x^{2} from both sides.

$$225-30x+145x^{2}-144x^{2}=0$$

Combine 145x^{2} and -144x^{2} to get x^{2}.

$$225-30x+x^{2}=0$$

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

$$x^{2}-30x+225=0$$

To solve the equation, factor x^{2}-30x+225 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-30 ab=225$$

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 225.

$$-1,-225 -3,-75 -5,-45 -9,-25 -15,-15$$

Calculate the sum for each pair.

$$-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30$$

The solution is the pair that gives sum -30.

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

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-15\right)\left(x-15\right)$$

Rewrite as a binomial square.

$$\left(x-15\right)^{2}$$

To find equation solution, solve x-15=0.

$$x=15$$

Steps Using Factoring By Grouping

Multiply 1 and 29 to get 29.

$$29\left(9-6x+5x^{2}\right)=36-144x+144x^{2}$$

Use the distributive property to multiply 29 by 9-6x+5x^{2}.

$$261-174x+145x^{2}=36-144x+144x^{2}$$

Subtract 36 from both sides.

$$261-174x+145x^{2}-36=-144x+144x^{2}$$

Subtract 36 from 261 to get 225.

$$225-174x+145x^{2}=-144x+144x^{2}$$

Add 144x to both sides.

$$225-174x+145x^{2}+144x=144x^{2}$$

Combine -174x and 144x to get -30x.

$$225-30x+145x^{2}=144x^{2}$$

Subtract 144x^{2} from both sides.

$$225-30x+145x^{2}-144x^{2}=0$$

Combine 145x^{2} and -144x^{2} to get x^{2}.

$$225-30x+x^{2}=0$$

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

$$x^{2}-30x+225=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+225. To find a and b, set up a system to be solved.

$$a+b=-30 ab=1\times 225=225$$

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 225.

$$-1,-225 -3,-75 -5,-45 -9,-25 -15,-15$$

Calculate the sum for each pair.

$$-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30$$

The solution is the pair that gives sum -30.

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

Rewrite x^{2}-30x+225 as \left(x^{2}-15x\right)+\left(-15x+225\right).

$$\left(x^{2}-15x\right)+\left(-15x+225\right)$$

Factor out x in the first and -15 in the second group.

$$x\left(x-15\right)-15\left(x-15\right)$$

Factor out common term x-15 by using distributive property.

$$\left(x-15\right)\left(x-15\right)$$

Rewrite as a binomial square.

$$\left(x-15\right)^{2}$$

To find equation solution, solve x-15=0.

$$x=15$$

Steps Using the Quadratic Formula

Multiply 1 and 29 to get 29.

$$29\left(9-6x+5x^{2}\right)=36-144x+144x^{2}$$

Use the distributive property to multiply 29 by 9-6x+5x^{2}.

$$261-174x+145x^{2}=36-144x+144x^{2}$$

Subtract 36 from both sides.

$$261-174x+145x^{2}-36=-144x+144x^{2}$$

Subtract 36 from 261 to get 225.

$$225-174x+145x^{2}=-144x+144x^{2}$$

Add 144x to both sides.

$$225-174x+145x^{2}+144x=144x^{2}$$

Combine -174x and 144x to get -30x.

$$225-30x+145x^{2}=144x^{2}$$

Subtract 144x^{2} from both sides.

$$225-30x+145x^{2}-144x^{2}=0$$

Combine 145x^{2} and -144x^{2} to get x^{2}.

$$225-30x+x^{2}=0$$

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.

$$x^{2}-30x+225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -30 for b, and 225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 225}}{2}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 225}}{2}$$

Multiply -4 times 225.

$$x=\frac{-\left(-30\right)±\sqrt{900-900}}{2}$$

Add 900 to -900.

$$x=\frac{-\left(-30\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{-30}{2}$$

The opposite of -30 is 30.

$$x=\frac{30}{2}$$

Divide 30 by 2.

$$x=15$$

Steps for Completing the Square

Multiply 1 and 29 to get 29.

$$29\left(9-6x+5x^{2}\right)=36-144x+144x^{2}$$

Use the distributive property to multiply 29 by 9-6x+5x^{2}.

$$261-174x+145x^{2}=36-144x+144x^{2}$$

Add 144x to both sides.

$$261-174x+145x^{2}+144x=36+144x^{2}$$

Combine -174x and 144x to get -30x.

$$261-30x+145x^{2}=36+144x^{2}$$

Subtract 144x^{2} from both sides.

$$261-30x+145x^{2}-144x^{2}=36$$

Combine 145x^{2} and -144x^{2} to get x^{2}.

$$261-30x+x^{2}=36$$

Subtract 261 from both sides.

$$-30x+x^{2}=36-261$$

Subtract 261 from 36 to get -225.

$$-30x+x^{2}=-225$$

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.

$$x^{2}-30x=-225$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-225+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-225+225$$

Add -225 to 225.

$$x^{2}-30x+225=0$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-15=0 x-15=0$$

Add 15 to both sides of the equation.

$$x=15 x=15$$

The equation is now solved. Solutions are the same.

$$x=15$$