Solve x-16x^2+49x-98=0

$$x - 16 x ^ { 2 } + 49 x - 98 = 0$$

Solve for x (complex solution)

$$x=\frac{25+\sqrt{943}i}{16}\approx 1.5625+1.919269067i$$

$$x=\frac{-\sqrt{943}i+25}{16}\approx 1.5625-1.919269067i$$

Steps Using the Quadratic Formula

Combine x and 49x to get 50x.

$$50x-16x^{2}-98=0$$

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.

$$-16x^{2}+50x-98=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 50 for b, and -98 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-16\right)\left(-98\right)}}{2\left(-16\right)}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\left(-16\right)\left(-98\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-50±\sqrt{2500+64\left(-98\right)}}{2\left(-16\right)}$$

Multiply 64 times -98.

$$x=\frac{-50±\sqrt{2500-6272}}{2\left(-16\right)}$$

Add 2500 to -6272.

$$x=\frac{-50±\sqrt{-3772}}{2\left(-16\right)}$$

Take the square root of -3772.

$$x=\frac{-50±2\sqrt{943}i}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-50±2\sqrt{943}i}{-32}$$

Now solve the equation x=\frac{-50±2\sqrt{943}i}{-32} when ± is plus. Add -50 to 2i\sqrt{943}.

$$x=\frac{-50+2\sqrt{943}i}{-32}$$

Divide -50+2i\sqrt{943} by -32.

$$x=\frac{-\sqrt{943}i+25}{16}$$

Now solve the equation x=\frac{-50±2\sqrt{943}i}{-32} when ± is minus. Subtract 2i\sqrt{943} from -50.

$$x=\frac{-2\sqrt{943}i-50}{-32}$$

Divide -50-2i\sqrt{943} by -32.

$$x=\frac{25+\sqrt{943}i}{16}$$

The equation is now solved.

$$x=\frac{-\sqrt{943}i+25}{16} x=\frac{25+\sqrt{943}i}{16}$$

Steps for Completing the Square

Combine x and 49x to get 50x.

$$50x-16x^{2}-98=0$$

Add 98 to both sides. Anything plus zero gives itself.

$$50x-16x^{2}=98$$

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.

$$-16x^{2}+50x=98$$

Divide both sides by -16.

$$\frac{-16x^{2}+50x}{-16}=\frac{98}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{50}{-16}x=\frac{98}{-16}$$

Reduce the fraction \frac{50}{-16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{25}{8}x=\frac{98}{-16}$$

Reduce the fraction \frac{98}{-16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{25}{8}x=-\frac{49}{8}$$

Divide -\frac{25}{8}, the coefficient of the x term, by 2 to get -\frac{25}{16}. Then add the square of -\frac{25}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{8}x+\left(-\frac{25}{16}\right)^{2}=-\frac{49}{8}+\left(-\frac{25}{16}\right)^{2}$$

Square -\frac{25}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{8}x+\frac{625}{256}=-\frac{49}{8}+\frac{625}{256}$$

Add -\frac{49}{8} to \frac{625}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{8}x+\frac{625}{256}=-\frac{943}{256}$$

Factor x^{2}-\frac{25}{8}x+\frac{625}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{16}\right)^{2}=-\frac{943}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{16}\right)^{2}}=\sqrt{-\frac{943}{256}}$$

Simplify.

$$x-\frac{25}{16}=\frac{\sqrt{943}i}{16} x-\frac{25}{16}=-\frac{\sqrt{943}i}{16}$$

Add \frac{25}{16} to both sides of the equation.

$$x=\frac{25+\sqrt{943}i}{16} x=\frac{-\sqrt{943}i+25}{16}$$