Solve (6-x*x)+(7*x)=30

$$(6-x \times x)+(7 \times x)=30$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{47}i+7}{2}\approx 3.5-3.4278273i$$

$$x=\frac{7+\sqrt{47}i}{2}\approx 3.5+3.4278273i$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$6-x^{2}+7x=30$$

Subtract 30 from both sides.

$$6-x^{2}+7x-30=0$$

Subtract 30 from 6 to get -24.

$$-24-x^{2}+7x=0$$

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.

$$-x^{2}+7x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\left(-24\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-7±\sqrt{49+4\left(-24\right)}}{2\left(-1\right)}$$

Multiply 4 times -24.

$$x=\frac{-7±\sqrt{49-96}}{2\left(-1\right)}$$

Add 49 to -96.

$$x=\frac{-7±\sqrt{-47}}{2\left(-1\right)}$$

Take the square root of -47.

$$x=\frac{-7±\sqrt{47}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-7±\sqrt{47}i}{-2}$$

Now solve the equation x=\frac{-7±\sqrt{47}i}{-2} when ± is plus. Add -7 to i\sqrt{47}.

$$x=\frac{-7+\sqrt{47}i}{-2}$$

Divide -7+i\sqrt{47} by -2.

$$x=\frac{-\sqrt{47}i+7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{47}i}{-2} when ± is minus. Subtract i\sqrt{47} from -7.

$$x=\frac{-\sqrt{47}i-7}{-2}$$

Divide -7-i\sqrt{47} by -2.

$$x=\frac{7+\sqrt{47}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{47}i+7}{2} x=\frac{7+\sqrt{47}i}{2}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$6-x^{2}+7x=30$$

Subtract 6 from both sides.

$$-x^{2}+7x=30-6$$

Subtract 6 from 30 to get 24.

$$-x^{2}+7x=24$$

Divide both sides by -1.

$$\frac{-x^{2}+7x}{-1}=\frac{24}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{7}{-1}x=\frac{24}{-1}$$

Divide 7 by -1.

$$x^{2}-7x=\frac{24}{-1}$$

Divide 24 by -1.

$$x^{2}-7x=-24$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-24+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-24+\frac{49}{4}$$

Add -24 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=-\frac{47}{4}$$

Factor x^{2}-7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=-\frac{47}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{47}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{47}i}{2} x-\frac{7}{2}=-\frac{\sqrt{47}i}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{7+\sqrt{47}i}{2} x=\frac{-\sqrt{47}i+7}{2}$$