Solve 7x^2+12=-5x

$$7 x ^ { 2 } + 12 = - 5 x$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{311}i}{14}\approx -0.357142857+1.259656578i$$

$$x=\frac{-\sqrt{311}i-5}{14}\approx -0.357142857-1.259656578i$$

Steps Using the Quadratic Formula

Add 5x to both sides.

$$7x^{2}+12+5x=0$$

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.

$$7x^{2}+5x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 5 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 7\times 12}}{2\times 7}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 7\times 12}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-5±\sqrt{25-28\times 12}}{2\times 7}$$

Multiply -28 times 12.

$$x=\frac{-5±\sqrt{25-336}}{2\times 7}$$

Add 25 to -336.

$$x=\frac{-5±\sqrt{-311}}{2\times 7}$$

Take the square root of -311.

$$x=\frac{-5±\sqrt{311}i}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-5±\sqrt{311}i}{14}$$

Now solve the equation x=\frac{-5±\sqrt{311}i}{14} when ± is plus. Add -5 to i\sqrt{311}.

$$x=\frac{-5+\sqrt{311}i}{14}$$

Now solve the equation x=\frac{-5±\sqrt{311}i}{14} when ± is minus. Subtract i\sqrt{311} from -5.

$$x=\frac{-\sqrt{311}i-5}{14}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{311}i}{14} x=\frac{-\sqrt{311}i-5}{14}$$

Steps for Completing the Square

Add 5x to both sides.

$$7x^{2}+12+5x=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$7x^{2}+5x=-12$$

Divide both sides by 7.

$$\frac{7x^{2}+5x}{7}=-\frac{12}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{5}{7}x=-\frac{12}{7}$$

Divide \frac{5}{7}, the coefficient of the x term, by 2 to get \frac{5}{14}. Then add the square of \frac{5}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{7}x+\left(\frac{5}{14}\right)^{2}=-\frac{12}{7}+\left(\frac{5}{14}\right)^{2}$$

Square \frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=-\frac{12}{7}+\frac{25}{196}$$

Add -\frac{12}{7} to \frac{25}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=-\frac{311}{196}$$

Factor x^{2}+\frac{5}{7}x+\frac{25}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{14}\right)^{2}=-\frac{311}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{14}\right)^{2}}=\sqrt{-\frac{311}{196}}$$

Simplify.

$$x+\frac{5}{14}=\frac{\sqrt{311}i}{14} x+\frac{5}{14}=-\frac{\sqrt{311}i}{14}$$

Subtract \frac{5}{14} from both sides of the equation.

$$x=\frac{-5+\sqrt{311}i}{14} x=\frac{-\sqrt{311}i-5}{14}$$