Solve 16k^2-40k=-23

$$16 k ^ { 2 } - 40 k = - 23$$

Solve for k

$$k = \frac{\sqrt{2} + 5}{4} \approx 1.603553391$$

$$k=\frac{5-\sqrt{2}}{4}\approx 0.896446609$$

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.

$$16k^{2}-40k=-23$$

Add 23 to both sides of the equation.

$$16k^{2}-40k-\left(-23\right)=-23-\left(-23\right)$$

Subtracting -23 from itself leaves 0.

$$16k^{2}-40k-\left(-23\right)=0$$

Subtract -23 from 0.

$$16k^{2}-40k+23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -40 for b, and 23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 16\times 23}}{2\times 16}$$

Square -40.

$$k=\frac{-\left(-40\right)±\sqrt{1600-4\times 16\times 23}}{2\times 16}$$

Multiply -4 times 16.

$$k=\frac{-\left(-40\right)±\sqrt{1600-64\times 23}}{2\times 16}$$

Multiply -64 times 23.

$$k=\frac{-\left(-40\right)±\sqrt{1600-1472}}{2\times 16}$$

Add 1600 to -1472.

$$k=\frac{-\left(-40\right)±\sqrt{128}}{2\times 16}$$

Take the square root of 128.

$$k=\frac{-\left(-40\right)±8\sqrt{2}}{2\times 16}$$

The opposite of -40 is 40.

$$k=\frac{40±8\sqrt{2}}{2\times 16}$$

Multiply 2 times 16.

$$k=\frac{40±8\sqrt{2}}{32}$$

Now solve the equation k=\frac{40±8\sqrt{2}}{32} when ± is plus. Add 40 to 8\sqrt{2}.

$$k=\frac{8\sqrt{2}+40}{32}$$

Divide 40+8\sqrt{2} by 32.

$$k=\frac{\sqrt{2}+5}{4}$$

Now solve the equation k=\frac{40±8\sqrt{2}}{32} when ± is minus. Subtract 8\sqrt{2} from 40.

$$k=\frac{40-8\sqrt{2}}{32}$$

Divide 40-8\sqrt{2} by 32.

$$k=\frac{5-\sqrt{2}}{4}$$

The equation is now solved.

$$k=\frac{\sqrt{2}+5}{4} k=\frac{5-\sqrt{2}}{4}$$

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.

$$16k^{2}-40k=-23$$

Divide both sides by 16.

$$\frac{16k^{2}-40k}{16}=-\frac{23}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$k^{2}+\left(-\frac{40}{16}\right)k=-\frac{23}{16}$$

Reduce the fraction \frac{-40}{16} to lowest terms by extracting and canceling out 8.

$$k^{2}-\frac{5}{2}k=-\frac{23}{16}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-\frac{5}{2}k+\left(-\frac{5}{4}\right)^{2}=-\frac{23}{16}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$k^{2}-\frac{5}{2}k+\frac{25}{16}=\frac{-23+25}{16}$$

Add -\frac{23}{16} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$k^{2}-\frac{5}{2}k+\frac{25}{16}=\frac{1}{8}$$

Factor k^{2}-\frac{5}{2}k+\frac{25}{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}.

$$\left(k-\frac{5}{4}\right)^{2}=\frac{1}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-\frac{5}{4}\right)^{2}}=\sqrt{\frac{1}{8}}$$

Simplify.

$$k-\frac{5}{4}=\frac{\sqrt{2}}{4} k-\frac{5}{4}=-\frac{\sqrt{2}}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$k=\frac{\sqrt{2}+5}{4} k=\frac{5-\sqrt{2}}{4}$$