Solve 20x^2-164=56x^2-59x-36

$$20 { x }^{ 2 } -164=56 { x }^{ 2 } -59x-36$$

Solve for x (complex solution)

$$x=\frac{59+\sqrt{14951}i}{72}\approx 0.819444444+1.698253914i$$

$$x=\frac{-\sqrt{14951}i+59}{72}\approx 0.819444444-1.698253914i$$

Steps Using the Quadratic Formula

Subtract 56x^{2} from both sides.

$$20x^{2}-164-56x^{2}=-59x-36$$

Combine 20x^{2} and -56x^{2} to get -36x^{2}.

$$-36x^{2}-164=-59x-36$$

Add 59x to both sides.

$$-36x^{2}-164+59x=-36$$

Add 36 to both sides.

$$-36x^{2}-164+59x+36=0$$

Add -164 and 36 to get -128.

$$-36x^{2}-128+59x=0$$

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.

$$-36x^{2}+59x-128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -36 for a, 59 for b, and -128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-59±\sqrt{59^{2}-4\left(-36\right)\left(-128\right)}}{2\left(-36\right)}$$

Square 59.

$$x=\frac{-59±\sqrt{3481-4\left(-36\right)\left(-128\right)}}{2\left(-36\right)}$$

Multiply -4 times -36.

$$x=\frac{-59±\sqrt{3481+144\left(-128\right)}}{2\left(-36\right)}$$

Multiply 144 times -128.

$$x=\frac{-59±\sqrt{3481-18432}}{2\left(-36\right)}$$

Add 3481 to -18432.

$$x=\frac{-59±\sqrt{-14951}}{2\left(-36\right)}$$

Take the square root of -14951.

$$x=\frac{-59±\sqrt{14951}i}{2\left(-36\right)}$$

Multiply 2 times -36.

$$x=\frac{-59±\sqrt{14951}i}{-72}$$

Now solve the equation x=\frac{-59±\sqrt{14951}i}{-72} when ± is plus. Add -59 to i\sqrt{14951}.

$$x=\frac{-59+\sqrt{14951}i}{-72}$$

Divide -59+i\sqrt{14951} by -72.

$$x=\frac{-\sqrt{14951}i+59}{72}$$

Now solve the equation x=\frac{-59±\sqrt{14951}i}{-72} when ± is minus. Subtract i\sqrt{14951} from -59.

$$x=\frac{-\sqrt{14951}i-59}{-72}$$

Divide -59-i\sqrt{14951} by -72.

$$x=\frac{59+\sqrt{14951}i}{72}$$

The equation is now solved.

$$x=\frac{-\sqrt{14951}i+59}{72} x=\frac{59+\sqrt{14951}i}{72}$$

Steps for Completing the Square

Subtract 56x^{2} from both sides.

$$20x^{2}-164-56x^{2}=-59x-36$$

Combine 20x^{2} and -56x^{2} to get -36x^{2}.

$$-36x^{2}-164=-59x-36$$

Add 59x to both sides.

$$-36x^{2}-164+59x=-36$$

Add 164 to both sides.

$$-36x^{2}+59x=-36+164$$

Add -36 and 164 to get 128.

$$-36x^{2}+59x=128$$

Divide both sides by -36.

$$\frac{-36x^{2}+59x}{-36}=\frac{128}{-36}$$

Dividing by -36 undoes the multiplication by -36.

$$x^{2}+\frac{59}{-36}x=\frac{128}{-36}$$

Divide 59 by -36.

$$x^{2}-\frac{59}{36}x=\frac{128}{-36}$$

Reduce the fraction \frac{128}{-36} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{59}{36}x=-\frac{32}{9}$$

Divide -\frac{59}{36}, the coefficient of the x term, by 2 to get -\frac{59}{72}. Then add the square of -\frac{59}{72} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{59}{36}x+\left(-\frac{59}{72}\right)^{2}=-\frac{32}{9}+\left(-\frac{59}{72}\right)^{2}$$

Square -\frac{59}{72} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{59}{36}x+\frac{3481}{5184}=-\frac{32}{9}+\frac{3481}{5184}$$

Add -\frac{32}{9} to \frac{3481}{5184} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{59}{36}x+\frac{3481}{5184}=-\frac{14951}{5184}$$

Factor x^{2}-\frac{59}{36}x+\frac{3481}{5184}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{59}{72}\right)^{2}=-\frac{14951}{5184}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{59}{72}\right)^{2}}=\sqrt{-\frac{14951}{5184}}$$

Simplify.

$$x-\frac{59}{72}=\frac{\sqrt{14951}i}{72} x-\frac{59}{72}=-\frac{\sqrt{14951}i}{72}$$

Add \frac{59}{72} to both sides of the equation.

$$x=\frac{59+\sqrt{14951}i}{72} x=\frac{-\sqrt{14951}i+59}{72}$$