Solve 0=x^2+900-60x

$$0 = x ^ { 2 } + 900 - 60 x$$

Solve for x

$$x=30$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+900-60x=0$$

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

$$x^{2}-60x+900=0$$

To solve the equation, factor x^{2}-60x+900 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.

$$a+b=-60 ab=900$$

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 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Calculate the sum for each pair.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

The solution is the pair that gives sum -60.

$$a=-30 b=-30$$

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

$$\left(x-30\right)\left(x-30\right)$$

Rewrite as a binomial square.

$$\left(x-30\right)^{2}$$

To find equation solution, solve x-30=0.

$$x=30$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+900-60x=0$$

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

$$x^{2}-60x+900=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+900. To find a and b, set up a system to be solved.

$$a+b=-60 ab=1\times 900=900$$

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 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Calculate the sum for each pair.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

The solution is the pair that gives sum -60.

$$a=-30 b=-30$$

Rewrite x^{2}-60x+900 as \left(x^{2}-30x\right)+\left(-30x+900\right).

$$\left(x^{2}-30x\right)+\left(-30x+900\right)$$

Factor out x in the first and -30 in the second group.

$$x\left(x-30\right)-30\left(x-30\right)$$

Factor out common term x-30 by using distributive property.

$$\left(x-30\right)\left(x-30\right)$$

Rewrite as a binomial square.

$$\left(x-30\right)^{2}$$

To find equation solution, solve x-30=0.

$$x=30$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+900-60x=0$$

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.

$$x^{2}-60x+900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -60 for b, and 900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 900}}{2}$$

Square -60.

$$x=\frac{-\left(-60\right)±\sqrt{3600-4\times 900}}{2}$$

Multiply -4 times 900.

$$x=\frac{-\left(-60\right)±\sqrt{3600-3600}}{2}$$

Add 3600 to -3600.

$$x=\frac{-\left(-60\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{-60}{2}$$

The opposite of -60 is 60.

$$x=\frac{60}{2}$$

Divide 60 by 2.

$$x=30$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+900-60x=0$$

Subtract 900 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-60x=-900$$

Divide -60, the coefficient of the x term, by 2 to get -30. Then add the square of -30 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-60x+\left(-30\right)^{2}=-900+\left(-30\right)^{2}$$

Square -30.

$$x^{2}-60x+900=-900+900$$

Add -900 to 900.

$$x^{2}-60x+900=0$$

Factor x^{2}-60x+900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-30\right)^{2}=0$$

Take the square root of both sides of the equation.

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

Simplify.

$$x-30=0 x-30=0$$

Add 30 to both sides of the equation.

$$x=30 x=30$$

The equation is now solved. Solutions are the same.

$$x=30$$