Solve 16x^2=56x-51

$$16 x ^ { 2 } = 56 x - 51$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{2}i}{4}\approx 1.75+0.353553391i$$

$$x=\frac{-\sqrt{2}i+7}{4}\approx 1.75-0.353553391i$$

Steps Using the Quadratic Formula

Subtract 56x from both sides.

$$16x^{2}-56x=-51$$

Add 51 to both sides.

$$16x^{2}-56x+51=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -56 for b, and 51 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 16\times 51}}{2\times 16}$$

Square -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 16\times 51}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-56\right)±\sqrt{3136-64\times 51}}{2\times 16}$$

Multiply -64 times 51.

$$x=\frac{-\left(-56\right)±\sqrt{3136-3264}}{2\times 16}$$

Add 3136 to -3264.

$$x=\frac{-\left(-56\right)±\sqrt{-128}}{2\times 16}$$

Take the square root of -128.

$$x=\frac{-\left(-56\right)±8\sqrt{2}i}{2\times 16}$$

The opposite of -56 is 56.

$$x=\frac{56±8\sqrt{2}i}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{56±8\sqrt{2}i}{32}$$

Now solve the equation x=\frac{56±8\sqrt{2}i}{32} when ± is plus. Add 56 to 8i\sqrt{2}.

$$x=\frac{56+2^{\frac{7}{2}}i}{32}$$

Divide 56+i\times 2^{\frac{7}{2}} by 32.

$$x=\frac{7+\sqrt{2}i}{4}$$

Now solve the equation x=\frac{56±8\sqrt{2}i}{32} when ± is minus. Subtract 8i\sqrt{2} from 56.

$$x=\frac{-2^{\frac{7}{2}}i+56}{32}$$

Divide 56-i\times 2^{\frac{7}{2}} by 32.

$$x=\frac{-\sqrt{2}i+7}{4}$$

The equation is now solved.

$$x=\frac{7+\sqrt{2}i}{4} x=\frac{-\sqrt{2}i+7}{4}$$

Steps for Completing the Square

Subtract 56x from both sides.

$$16x^{2}-56x=-51$$

Divide both sides by 16.

$$\frac{16x^{2}-56x}{16}=-\frac{51}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{56}{16}\right)x=-\frac{51}{16}$$

Reduce the fraction \frac{-56}{16} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{7}{2}x=-\frac{51}{16}$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-\frac{51}{16}+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{-51+49}{16}$$

Add -\frac{51}{16} to \frac{49}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-\frac{1}{8}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=-\frac{1}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{-\frac{1}{8}}$$

Simplify.

$$x-\frac{7}{4}=\frac{\sqrt{2}i}{4} x-\frac{7}{4}=-\frac{\sqrt{2}i}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=\frac{7+\sqrt{2}i}{4} x=\frac{-\sqrt{2}i+7}{4}$$