Solve 0=141-21x-16x^2

$$0=141-21x-16 { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{9465} - 21}{32} \approx 2.384007236$$

$$x=\frac{-\sqrt{9465}-21}{32}\approx -3.696507236$$

Steps Using the Quadratic Formula

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

$$141-21x-16x^{2}=0$$

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.

$$-16x^{2}-21x+141=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, -21 for b, and 141 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-16\right)\times 141}}{2\left(-16\right)}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\left(-16\right)\times 141}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-\left(-21\right)±\sqrt{441+64\times 141}}{2\left(-16\right)}$$

Multiply 64 times 141.

$$x=\frac{-\left(-21\right)±\sqrt{441+9024}}{2\left(-16\right)}$$

Add 441 to 9024.

$$x=\frac{-\left(-21\right)±\sqrt{9465}}{2\left(-16\right)}$$

The opposite of -21 is 21.

$$x=\frac{21±\sqrt{9465}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{21±\sqrt{9465}}{-32}$$

Now solve the equation x=\frac{21±\sqrt{9465}}{-32} when ± is plus. Add 21 to \sqrt{9465}.

$$x=\frac{\sqrt{9465}+21}{-32}$$

Divide 21+\sqrt{9465} by -32.

$$x=\frac{-\sqrt{9465}-21}{32}$$

Now solve the equation x=\frac{21±\sqrt{9465}}{-32} when ± is minus. Subtract \sqrt{9465} from 21.

$$x=\frac{21-\sqrt{9465}}{-32}$$

Divide 21-\sqrt{9465} by -32.

$$x=\frac{\sqrt{9465}-21}{32}$$

The equation is now solved.

$$x=\frac{-\sqrt{9465}-21}{32} x=\frac{\sqrt{9465}-21}{32}$$

Steps for Completing the Square

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

$$141-21x-16x^{2}=0$$

Subtract 141 from both sides. Anything subtracted from zero gives its negation.

$$-21x-16x^{2}=-141$$

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.

$$-16x^{2}-21x=-141$$

Divide both sides by -16.

$$\frac{-16x^{2}-21x}{-16}=-\frac{141}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\left(-\frac{21}{-16}\right)x=-\frac{141}{-16}$$

Divide -21 by -16.

$$x^{2}+\frac{21}{16}x=-\frac{141}{-16}$$

Divide -141 by -16.

$$x^{2}+\frac{21}{16}x=\frac{141}{16}$$

Divide \frac{21}{16}, the coefficient of the x term, by 2 to get \frac{21}{32}. Then add the square of \frac{21}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{21}{16}x+\left(\frac{21}{32}\right)^{2}=\frac{141}{16}+\left(\frac{21}{32}\right)^{2}$$

Square \frac{21}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{21}{16}x+\frac{441}{1024}=\frac{141}{16}+\frac{441}{1024}$$

Add \frac{141}{16} to \frac{441}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{21}{16}x+\frac{441}{1024}=\frac{9465}{1024}$$

Factor x^{2}+\frac{21}{16}x+\frac{441}{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}.

$$\left(x+\frac{21}{32}\right)^{2}=\frac{9465}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{21}{32}\right)^{2}}=\sqrt{\frac{9465}{1024}}$$

Simplify.

$$x+\frac{21}{32}=\frac{\sqrt{9465}}{32} x+\frac{21}{32}=-\frac{\sqrt{9465}}{32}$$

Subtract \frac{21}{32} from both sides of the equation.

$$x=\frac{\sqrt{9465}-21}{32} x=\frac{-\sqrt{9465}-21}{32}$$