Solve -21x^2+77x=-30+18x

$$- 21 x ^ { 2 } + 77 x = - 30 + 18 x$$

Solve for x

$$x = \frac{\sqrt{6001} + 59}{42} \approx 3.249193372$$

$$x=\frac{59-\sqrt{6001}}{42}\approx -0.439669563$$

Steps Using the Quadratic Formula

Subtract -30 from both sides.

$$-21x^{2}+77x-\left(-30\right)=18x$$

The opposite of -30 is 30.

$$-21x^{2}+77x+30=18x$$

Subtract 18x from both sides.

$$-21x^{2}+77x+30-18x=0$$

Combine 77x and -18x to get 59x.

$$-21x^{2}+59x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -21 for a, 59 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-59±\sqrt{59^{2}-4\left(-21\right)\times 30}}{2\left(-21\right)}$$

Square 59.

$$x=\frac{-59±\sqrt{3481-4\left(-21\right)\times 30}}{2\left(-21\right)}$$

Multiply -4 times -21.

$$x=\frac{-59±\sqrt{3481+84\times 30}}{2\left(-21\right)}$$

Multiply 84 times 30.

$$x=\frac{-59±\sqrt{3481+2520}}{2\left(-21\right)}$$

Add 3481 to 2520.

$$x=\frac{-59±\sqrt{6001}}{2\left(-21\right)}$$

Multiply 2 times -21.

$$x=\frac{-59±\sqrt{6001}}{-42}$$

Now solve the equation x=\frac{-59±\sqrt{6001}}{-42} when ± is plus. Add -59 to \sqrt{6001}.

$$x=\frac{\sqrt{6001}-59}{-42}$$

Divide -59+\sqrt{6001} by -42.

$$x=\frac{59-\sqrt{6001}}{42}$$

Now solve the equation x=\frac{-59±\sqrt{6001}}{-42} when ± is minus. Subtract \sqrt{6001} from -59.

$$x=\frac{-\sqrt{6001}-59}{-42}$$

Divide -59-\sqrt{6001} by -42.

$$x=\frac{\sqrt{6001}+59}{42}$$

The equation is now solved.

$$x=\frac{59-\sqrt{6001}}{42} x=\frac{\sqrt{6001}+59}{42}$$

Steps for Completing the Square

Subtract 18x from both sides.

$$-21x^{2}+77x-18x=-30$$

Combine 77x and -18x to get 59x.

$$-21x^{2}+59x=-30$$

Divide both sides by -21.

$$\frac{-21x^{2}+59x}{-21}=-\frac{30}{-21}$$

Dividing by -21 undoes the multiplication by -21.

$$x^{2}+\frac{59}{-21}x=-\frac{30}{-21}$$

Divide 59 by -21.

$$x^{2}-\frac{59}{21}x=-\frac{30}{-21}$$

Reduce the fraction \frac{-30}{-21} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{59}{21}x=\frac{10}{7}$$

Divide -\frac{59}{21}, the coefficient of the x term, by 2 to get -\frac{59}{42}. Then add the square of -\frac{59}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{59}{21}x+\left(-\frac{59}{42}\right)^{2}=\frac{10}{7}+\left(-\frac{59}{42}\right)^{2}$$

Square -\frac{59}{42} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{59}{21}x+\frac{3481}{1764}=\frac{10}{7}+\frac{3481}{1764}$$

Add \frac{10}{7} to \frac{3481}{1764} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{59}{21}x+\frac{3481}{1764}=\frac{6001}{1764}$$

Factor x^{2}-\frac{59}{21}x+\frac{3481}{1764}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{59}{42}\right)^{2}=\frac{6001}{1764}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{59}{42}\right)^{2}}=\sqrt{\frac{6001}{1764}}$$

Simplify.

$$x-\frac{59}{42}=\frac{\sqrt{6001}}{42} x-\frac{59}{42}=-\frac{\sqrt{6001}}{42}$$

Add \frac{59}{42} to both sides of the equation.

$$x=\frac{\sqrt{6001}+59}{42} x=\frac{59-\sqrt{6001}}{42}$$