Solve 288x-1152=30+x^2-9x+20

$$288 x - 1152 = 30 + x ^ { 2 } - 9 x + 20$$

Solve for x

$$x = \frac{\sqrt{83401} + 297}{2} \approx 292.896156459$$

$$x = \frac{297 - \sqrt{83401}}{2} \approx 4.103843541$$

Steps Using the Quadratic Formula

Add 30 and 20 to get 50.

$$288x-1152=50+x^{2}-9x$$

Subtract 50 from both sides.

$$288x-1152-50=x^{2}-9x$$

Subtract 50 from -1152 to get -1202.

$$288x-1202=x^{2}-9x$$

Subtract x^{2} from both sides.

$$288x-1202-x^{2}=-9x$$

Add 9x to both sides.

$$288x-1202-x^{2}+9x=0$$

Combine 288x and 9x to get 297x.

$$297x-1202-x^{2}=0$$

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.

$$-x^{2}+297x-1202=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 297 for b, and -1202 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-297±\sqrt{297^{2}-4\left(-1\right)\left(-1202\right)}}{2\left(-1\right)}$$

Square 297.

$$x=\frac{-297±\sqrt{88209-4\left(-1\right)\left(-1202\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-297±\sqrt{88209+4\left(-1202\right)}}{2\left(-1\right)}$$

Multiply 4 times -1202.

$$x=\frac{-297±\sqrt{88209-4808}}{2\left(-1\right)}$$

Add 88209 to -4808.

$$x=\frac{-297±\sqrt{83401}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-297±\sqrt{83401}}{-2}$$

Now solve the equation x=\frac{-297±\sqrt{83401}}{-2} when ± is plus. Add -297 to \sqrt{83401}.

$$x=\frac{\sqrt{83401}-297}{-2}$$

Divide -297+\sqrt{83401} by -2.

$$x=\frac{297-\sqrt{83401}}{2}$$

Now solve the equation x=\frac{-297±\sqrt{83401}}{-2} when ± is minus. Subtract \sqrt{83401} from -297.

$$x=\frac{-\sqrt{83401}-297}{-2}$$

Divide -297-\sqrt{83401} by -2.

$$x=\frac{\sqrt{83401}+297}{2}$$

The equation is now solved.

$$x=\frac{297-\sqrt{83401}}{2} x=\frac{\sqrt{83401}+297}{2}$$

Steps for Completing the Square

Add 30 and 20 to get 50.

$$288x-1152=50+x^{2}-9x$$

Subtract x^{2} from both sides.

$$288x-1152-x^{2}=50-9x$$

Add 9x to both sides.

$$288x-1152-x^{2}+9x=50$$

Combine 288x and 9x to get 297x.

$$297x-1152-x^{2}=50$$

Add 1152 to both sides.

$$297x-x^{2}=50+1152$$

Add 50 and 1152 to get 1202.

$$297x-x^{2}=1202$$

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.

$$-x^{2}+297x=1202$$

Divide both sides by -1.

$$\frac{-x^{2}+297x}{-1}=\frac{1202}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{297}{-1}x=\frac{1202}{-1}$$

Divide 297 by -1.

$$x^{2}-297x=\frac{1202}{-1}$$

Divide 1202 by -1.

$$x^{2}-297x=-1202$$

Divide -297, the coefficient of the x term, by 2 to get -\frac{297}{2}. Then add the square of -\frac{297}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-297x+\left(-\frac{297}{2}\right)^{2}=-1202+\left(-\frac{297}{2}\right)^{2}$$

Square -\frac{297}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-297x+\frac{88209}{4}=-1202+\frac{88209}{4}$$

Add -1202 to \frac{88209}{4}.

$$x^{2}-297x+\frac{88209}{4}=\frac{83401}{4}$$

Factor x^{2}-297x+\frac{88209}{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}.

$$\left(x-\frac{297}{2}\right)^{2}=\frac{83401}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{297}{2}\right)^{2}}=\sqrt{\frac{83401}{4}}$$

Simplify.

$$x-\frac{297}{2}=\frac{\sqrt{83401}}{2} x-\frac{297}{2}=-\frac{\sqrt{83401}}{2}$$

Add \frac{297}{2} to both sides of the equation.

$$x=\frac{\sqrt{83401}+297}{2} x=\frac{297-\sqrt{83401}}{2}$$