Solve 16x^2-24x=1

$$16 x ^ { 2 } - 24 x = 1$$

Solve for x

$$x = \frac{\sqrt{10} + 3}{4} \approx 1.540569415$$

$$x=\frac{3-\sqrt{10}}{4}\approx -0.040569415$$

Steps Using the Quadratic Formula

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.

$$16x^{2}-24x=1$$

Subtract 1 from both sides of the equation.

$$16x^{2}-24x-1=1-1$$

Subtracting 1 from itself leaves 0.

$$16x^{2}-24x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -24 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 16\left(-1\right)}}{2\times 16}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 16\left(-1\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-24\right)±\sqrt{576-64\left(-1\right)}}{2\times 16}$$

Multiply -64 times -1.

$$x=\frac{-\left(-24\right)±\sqrt{576+64}}{2\times 16}$$

Add 576 to 64.

$$x=\frac{-\left(-24\right)±\sqrt{640}}{2\times 16}$$

Take the square root of 640.

$$x=\frac{-\left(-24\right)±8\sqrt{10}}{2\times 16}$$

The opposite of -24 is 24.

$$x=\frac{24±8\sqrt{10}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{24±8\sqrt{10}}{32}$$

Now solve the equation x=\frac{24±8\sqrt{10}}{32} when ± is plus. Add 24 to 8\sqrt{10}.

$$x=\frac{8\sqrt{10}+24}{32}$$

Divide 24+8\sqrt{10} by 32.

$$x=\frac{\sqrt{10}+3}{4}$$

Now solve the equation x=\frac{24±8\sqrt{10}}{32} when ± is minus. Subtract 8\sqrt{10} from 24.

$$x=\frac{24-8\sqrt{10}}{32}$$

Divide 24-8\sqrt{10} by 32.

$$x=\frac{3-\sqrt{10}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{10}+3}{4} x=\frac{3-\sqrt{10}}{4}$$

Steps for Completing the Square

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.

$$16x^{2}-24x=1$$

Divide both sides by 16.

$$\frac{16x^{2}-24x}{16}=\frac{1}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{24}{16}\right)x=\frac{1}{16}$$

Reduce the fraction \frac{-24}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{3}{2}x=\frac{1}{16}$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{1}{16}+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{1+9}{16}$$

Add \frac{1}{16} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{5}{8}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{5}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{5}{8}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{10}}{4} x-\frac{3}{4}=-\frac{\sqrt{10}}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{\sqrt{10}+3}{4} x=\frac{3-\sqrt{10}}{4}$$