Solve 3-12+16x+9+16x^2-24x=0

$$3 - 12 + 16 x + 9 + 16 x ^ { 2 } - 24 x = 0$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=0$$

Steps Using Factoring

Subtract 12 from 3 to get -9.

$$-9+16x+9+16x^{2}-24x=0$$

Add -9 and 9 to get 0.

$$16x+16x^{2}-24x=0$$

Combine 16x and -24x to get -8x.

$$-8x+16x^{2}=0$$

Factor out x.

$$x\left(-8+16x\right)=0$$

To find equation solutions, solve x=0 and -8+16x=0.

$$x=0 x=\frac{1}{2}$$

Steps Using the Quadratic Formula

Subtract 12 from 3 to get -9.

$$-9+16x+9+16x^{2}-24x=0$$

Add -9 and 9 to get 0.

$$16x+16x^{2}-24x=0$$

Combine 16x and -24x to get -8x.

$$-8x+16x^{2}=0$$

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.

$$16x^{2}-8x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -8 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}}}{2\times 16}$$

Take the square root of \left(-8\right)^{2}.

$$x=\frac{-\left(-8\right)±8}{2\times 16}$$

The opposite of -8 is 8.

$$x=\frac{8±8}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{8±8}{32}$$

Now solve the equation x=\frac{8±8}{32} when ± is plus. Add 8 to 8.

$$x=\frac{16}{32}$$

Reduce the fraction \frac{16}{32} to lowest terms by extracting and canceling out 16.

$$x=\frac{1}{2}$$

Now solve the equation x=\frac{8±8}{32} when ± is minus. Subtract 8 from 8.

$$x=\frac{0}{32}$$

Divide 0 by 32.

$$x=0$$

The equation is now solved.

$$x=\frac{1}{2} x=0$$

Steps for Completing the Square

Subtract 12 from 3 to get -9.

$$-9+16x+9+16x^{2}-24x=0$$

Add -9 and 9 to get 0.

$$16x+16x^{2}-24x=0$$

Combine 16x and -24x to get -8x.

$$-8x+16x^{2}=0$$

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.

$$16x^{2}-8x=0$$

Divide both sides by 16.

$$\frac{16x^{2}-8x}{16}=\frac{0}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{8}{16}\right)x=\frac{0}{16}$$

Reduce the fraction \frac{-8}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{1}{2}x=\frac{0}{16}$$

Divide 0 by 16.

$$x^{2}-\frac{1}{2}x=0$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{1}{4} x-\frac{1}{4}=-\frac{1}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1}{2} x=0$$