Solve 2x^2-28x=-134

$$2 x ^ { 2 } - 28 x = - 134$$

Solve for x (complex solution)

$$x=7+3\sqrt{2}i\approx 7+4.242640687i$$

$$x=-3\sqrt{2}i+7\approx 7-4.242640687i$$

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.

$$2x^{2}-28x=-134$$

Add 134 to both sides of the equation.

$$2x^{2}-28x-\left(-134\right)=-134-\left(-134\right)$$

Subtracting -134 from itself leaves 0.

$$2x^{2}-28x-\left(-134\right)=0$$

Subtract -134 from 0.

$$2x^{2}-28x+134=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -28 for b, and 134 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 134}}{2\times 2}$$

Square -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 2\times 134}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-28\right)±\sqrt{784-8\times 134}}{2\times 2}$$

Multiply -8 times 134.

$$x=\frac{-\left(-28\right)±\sqrt{784-1072}}{2\times 2}$$

Add 784 to -1072.

$$x=\frac{-\left(-28\right)±\sqrt{-288}}{2\times 2}$$

Take the square root of -288.

$$x=\frac{-\left(-28\right)±12\sqrt{2}i}{2\times 2}$$

The opposite of -28 is 28.

$$x=\frac{28±12\sqrt{2}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{28±12\sqrt{2}i}{4}$$

Now solve the equation x=\frac{28±12\sqrt{2}i}{4} when ± is plus. Add 28 to 12i\sqrt{2}.

$$x=\frac{28+3\times 2^{\frac{5}{2}}i}{4}$$

Divide 28+3i\times 2^{\frac{5}{2}} by 4.

$$x=7+3\sqrt{2}i$$

Now solve the equation x=\frac{28±12\sqrt{2}i}{4} when ± is minus. Subtract 12i\sqrt{2} from 28.

$$x=\frac{-3\times 2^{\frac{5}{2}}i+28}{4}$$

Divide 28-3i\times 2^{\frac{5}{2}} by 4.

$$x=-3\sqrt{2}i+7$$

The equation is now solved.

$$x=7+3\sqrt{2}i x=-3\sqrt{2}i+7$$

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.

$$2x^{2}-28x=-134$$

Divide both sides by 2.

$$\frac{2x^{2}-28x}{2}=-\frac{134}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{28}{2}\right)x=-\frac{134}{2}$$

Divide -28 by 2.

$$x^{2}-14x=-\frac{134}{2}$$

Divide -134 by 2.

$$x^{2}-14x=-67$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-67+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-67+49$$

Add -67 to 49.

$$x^{2}-14x+49=-18$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=-18$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{-18}$$

Simplify.

$$x-7=3\sqrt{2}i x-7=-3\sqrt{2}i$$

Add 7 to both sides of the equation.

$$x=7+3\sqrt{2}i x=-3\sqrt{2}i+7$$