Solve 21=3+35x-16x^2

$$21 = 3 + 35 x - 16 x ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{73} + 35}{32} \approx 1.360750117$$

$$x=\frac{35-\sqrt{73}}{32}\approx 0.826749883$$

Steps Using the Quadratic Formula

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

$$3+35x-16x^{2}=21$$

Subtract 21 from both sides.

$$3+35x-16x^{2}-21=0$$

Subtract 21 from 3 to get -18.

$$-18+35x-16x^{2}=0$$

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.

$$-16x^{2}+35x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 35 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-35±\sqrt{35^{2}-4\left(-16\right)\left(-18\right)}}{2\left(-16\right)}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\left(-16\right)\left(-18\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-35±\sqrt{1225+64\left(-18\right)}}{2\left(-16\right)}$$

Multiply 64 times -18.

$$x=\frac{-35±\sqrt{1225-1152}}{2\left(-16\right)}$$

Add 1225 to -1152.

$$x=\frac{-35±\sqrt{73}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-35±\sqrt{73}}{-32}$$

Now solve the equation x=\frac{-35±\sqrt{73}}{-32} when ± is plus. Add -35 to \sqrt{73}.

$$x=\frac{\sqrt{73}-35}{-32}$$

Divide -35+\sqrt{73} by -32.

$$x=\frac{35-\sqrt{73}}{32}$$

Now solve the equation x=\frac{-35±\sqrt{73}}{-32} when ± is minus. Subtract \sqrt{73} from -35.

$$x=\frac{-\sqrt{73}-35}{-32}$$

Divide -35-\sqrt{73} by -32.

$$x=\frac{\sqrt{73}+35}{32}$$

The equation is now solved.

$$x=\frac{35-\sqrt{73}}{32} x=\frac{\sqrt{73}+35}{32}$$

Steps for Completing the Square

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

$$3+35x-16x^{2}=21$$

Subtract 3 from both sides.

$$35x-16x^{2}=21-3$$

Subtract 3 from 21 to get 18.

$$35x-16x^{2}=18$$

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.

$$-16x^{2}+35x=18$$

Divide both sides by -16.

$$\frac{-16x^{2}+35x}{-16}=\frac{18}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{35}{-16}x=\frac{18}{-16}$$

Divide 35 by -16.

$$x^{2}-\frac{35}{16}x=\frac{18}{-16}$$

Reduce the fraction \frac{18}{-16} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{35}{16}x=-\frac{9}{8}$$

Divide -\frac{35}{16}, the coefficient of the x term, by 2 to get -\frac{35}{32}. Then add the square of -\frac{35}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{16}x+\left(-\frac{35}{32}\right)^{2}=-\frac{9}{8}+\left(-\frac{35}{32}\right)^{2}$$

Square -\frac{35}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{16}x+\frac{1225}{1024}=-\frac{9}{8}+\frac{1225}{1024}$$

Add -\frac{9}{8} to \frac{1225}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{35}{16}x+\frac{1225}{1024}=\frac{73}{1024}$$

Factor x^{2}-\frac{35}{16}x+\frac{1225}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{32}\right)^{2}=\frac{73}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{32}\right)^{2}}=\sqrt{\frac{73}{1024}}$$

Simplify.

$$x-\frac{35}{32}=\frac{\sqrt{73}}{32} x-\frac{35}{32}=-\frac{\sqrt{73}}{32}$$

Add \frac{35}{32} to both sides of the equation.

$$x=\frac{\sqrt{73}+35}{32} x=\frac{35-\sqrt{73}}{32}$$