Solve 54x-45-9x^2=0

$$54 x - 45 - 9 x ^ { 2 } = 0$$

Solve for x

$$x=1$$

$$x=5$$

Steps Using Factoring By Grouping

Divide both sides by 9.

$$6x-5-x^{2}=0$$

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

$$-x^{2}+6x-5=0$$

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

$$a+b=6 ab=-\left(-5\right)=5$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=5 b=1$$

Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).

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

Factor out -x in -x^{2}+5x.

$$-x\left(x-5\right)+x-5$$

Factor out common term x-5 by using distributive property.

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

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

$$x=5 x=1$$

Steps Using the Quadratic Formula

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.

$$-9x^{2}+54x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 54 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-54±\sqrt{54^{2}-4\left(-9\right)\left(-45\right)}}{2\left(-9\right)}$$

Square 54.

$$x=\frac{-54±\sqrt{2916-4\left(-9\right)\left(-45\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-54±\sqrt{2916+36\left(-45\right)}}{2\left(-9\right)}$$

Multiply 36 times -45.

$$x=\frac{-54±\sqrt{2916-1620}}{2\left(-9\right)}$$

Add 2916 to -1620.

$$x=\frac{-54±\sqrt{1296}}{2\left(-9\right)}$$

Take the square root of 1296.

$$x=\frac{-54±36}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-54±36}{-18}$$

Now solve the equation x=\frac{-54±36}{-18} when ± is plus. Add -54 to 36.

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

Divide -18 by -18.

$$x=1$$

Now solve the equation x=\frac{-54±36}{-18} when ± is minus. Subtract 36 from -54.

$$x=-\frac{90}{-18}$$

Divide -90 by -18.

$$x=5$$

The equation is now solved.

$$x=1 x=5$$

Steps for Completing the Square

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.

$$-9x^{2}+54x-45=0$$

Add 45 to both sides of the equation.

$$-9x^{2}+54x-45-\left(-45\right)=-\left(-45\right)$$

Subtracting -45 from itself leaves 0.

$$-9x^{2}+54x=-\left(-45\right)$$

Subtract -45 from 0.

$$-9x^{2}+54x=45$$

Divide both sides by -9.

$$\frac{-9x^{2}+54x}{-9}=\frac{45}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{54}{-9}x=\frac{45}{-9}$$

Divide 54 by -9.

$$x^{2}-6x=\frac{45}{-9}$$

Divide 45 by -9.

$$x^{2}-6x=-5$$

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

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

Square -3.

$$x^{2}-6x+9=-5+9$$

Add -5 to 9.

$$x^{2}-6x+9=4$$

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

$$\left(x-3\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-3=2 x-3=-2$$

Add 3 to both sides of the equation.

$$x=5 x=1$$