Solve 48x+52x-x^2=2300

$$48 x + 52 x - x ^ { 2 } = 2300$$

Solve for x

$$x=10\sqrt{2}+50\approx 64.142135624$$

$$x=50-10\sqrt{2}\approx 35.857864376$$

Steps Using the Quadratic Formula

Combine 48x and 52x to get 100x.

$$100x-x^{2}=2300$$

Subtract 2300 from both sides.

$$100x-x^{2}-2300=0$$

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.

$$-x^{2}+100x-2300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 100 for b, and -2300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-1\right)\left(-2300\right)}}{2\left(-1\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-1\right)\left(-2300\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-100±\sqrt{10000+4\left(-2300\right)}}{2\left(-1\right)}$$

Multiply 4 times -2300.

$$x=\frac{-100±\sqrt{10000-9200}}{2\left(-1\right)}$$

Add 10000 to -9200.

$$x=\frac{-100±\sqrt{800}}{2\left(-1\right)}$$

Take the square root of 800.

$$x=\frac{-100±20\sqrt{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-100±20\sqrt{2}}{-2}$$

Now solve the equation x=\frac{-100±20\sqrt{2}}{-2} when ± is plus. Add -100 to 20\sqrt{2}.

$$x=\frac{20\sqrt{2}-100}{-2}$$

Divide -100+20\sqrt{2} by -2.

$$x=50-10\sqrt{2}$$

Now solve the equation x=\frac{-100±20\sqrt{2}}{-2} when ± is minus. Subtract 20\sqrt{2} from -100.

$$x=\frac{-20\sqrt{2}-100}{-2}$$

Divide -100-20\sqrt{2} by -2.

$$x=10\sqrt{2}+50$$

The equation is now solved.

$$x=50-10\sqrt{2} x=10\sqrt{2}+50$$

Steps for Completing the Square

Combine 48x and 52x to get 100x.

$$100x-x^{2}=2300$$

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.

$$-x^{2}+100x=2300$$

Divide both sides by -1.

$$\frac{-x^{2}+100x}{-1}=\frac{2300}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{100}{-1}x=\frac{2300}{-1}$$

Divide 100 by -1.

$$x^{2}-100x=\frac{2300}{-1}$$

Divide 2300 by -1.

$$x^{2}-100x=-2300$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-2300+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-2300+2500$$

Add -2300 to 2500.

$$x^{2}-100x+2500=200$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=200$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{200}$$

Simplify.

$$x-50=10\sqrt{2} x-50=-10\sqrt{2}$$

Add 50 to both sides of the equation.

$$x=10\sqrt{2}+50 x=50-10\sqrt{2}$$