Solve 5x*10-9xdiv2x=99

$$5x \times 10-9x \div 2x=99$$

Solve for x

$$x = \frac{\sqrt{718} + 50}{9} \approx 8.532835779$$

$$x = \frac{50 - \sqrt{718}}{9} \approx 2.578275332$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$10x\times 10-9xx=198$$

Multiply 10 and 10 to get 100.

$$100x-9xx=198$$

Multiply x and x to get x^{2}.

$$100x-9x^{2}=198$$

Subtract 198 from both sides.

$$100x-9x^{2}-198=0$$

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.

$$-9x^{2}+100x-198=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 100 for b, and -198 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-9\right)\left(-198\right)}}{2\left(-9\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-9\right)\left(-198\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-100±\sqrt{10000+36\left(-198\right)}}{2\left(-9\right)}$$

Multiply 36 times -198.

$$x=\frac{-100±\sqrt{10000-7128}}{2\left(-9\right)}$$

Add 10000 to -7128.

$$x=\frac{-100±\sqrt{2872}}{2\left(-9\right)}$$

Take the square root of 2872.

$$x=\frac{-100±2\sqrt{718}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-100±2\sqrt{718}}{-18}$$

Now solve the equation x=\frac{-100±2\sqrt{718}}{-18} when ± is plus. Add -100 to 2\sqrt{718}.

$$x=\frac{2\sqrt{718}-100}{-18}$$

Divide -100+2\sqrt{718} by -18.

$$x=\frac{50-\sqrt{718}}{9}$$

Now solve the equation x=\frac{-100±2\sqrt{718}}{-18} when ± is minus. Subtract 2\sqrt{718} from -100.

$$x=\frac{-2\sqrt{718}-100}{-18}$$

Divide -100-2\sqrt{718} by -18.

$$x=\frac{\sqrt{718}+50}{9}$$

The equation is now solved.

$$x=\frac{50-\sqrt{718}}{9} x=\frac{\sqrt{718}+50}{9}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$10x\times 10-9xx=198$$

Multiply 10 and 10 to get 100.

$$100x-9xx=198$$

Multiply x and x to get x^{2}.

$$100x-9x^{2}=198$$

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.

$$-9x^{2}+100x=198$$

Divide both sides by -9.

$$\frac{-9x^{2}+100x}{-9}=\frac{198}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\frac{100}{-9}x=\frac{198}{-9}$$

Divide 100 by -9.

$$x^{2}-\frac{100}{9}x=\frac{198}{-9}$$

Divide 198 by -9.

$$x^{2}-\frac{100}{9}x=-22$$

Divide -\frac{100}{9}, the coefficient of the x term, by 2 to get -\frac{50}{9}. Then add the square of -\frac{50}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{100}{9}x+\left(-\frac{50}{9}\right)^{2}=-22+\left(-\frac{50}{9}\right)^{2}$$

Square -\frac{50}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{100}{9}x+\frac{2500}{81}=-22+\frac{2500}{81}$$

Add -22 to \frac{2500}{81}.

$$x^{2}-\frac{100}{9}x+\frac{2500}{81}=\frac{718}{81}$$

Factor x^{2}-\frac{100}{9}x+\frac{2500}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{50}{9}\right)^{2}=\frac{718}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{50}{9}\right)^{2}}=\sqrt{\frac{718}{81}}$$

Simplify.

$$x-\frac{50}{9}=\frac{\sqrt{718}}{9} x-\frac{50}{9}=-\frac{\sqrt{718}}{9}$$

Add \frac{50}{9} to both sides of the equation.

$$x=\frac{\sqrt{718}+50}{9} x=\frac{50-\sqrt{718}}{9}$$