Solve 150/4=49+x^2-11x

$$\frac { 150 } { 4 } = 49 + x ^ { 2 } - 11 x$$

Solve for x

$$x = \frac{5 \sqrt{3} + 11}{2} \approx 9.830127019$$

$$x = \frac{11 - 5 \sqrt{3}}{2} \approx 1.169872981$$

Steps Using the Quadratic Formula

Reduce the fraction \frac{150}{4} to lowest terms by extracting and canceling out 2.

$$\frac{75}{2}=49+x^{2}-11x$$

Swap sides so that all variable terms are on the left hand side.

$$49+x^{2}-11x=\frac{75}{2}$$

Subtract \frac{75}{2} from both sides.

$$49+x^{2}-11x-\frac{75}{2}=0$$

Subtract \frac{75}{2} from 49 to get \frac{23}{2}.

$$\frac{23}{2}+x^{2}-11x=0$$

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.

$$x^{2}-11x+\frac{23}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and \frac{23}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times \frac{23}{2}}}{2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times \frac{23}{2}}}{2}$$

Multiply -4 times \frac{23}{2}.

$$x=\frac{-\left(-11\right)±\sqrt{121-46}}{2}$$

Add 121 to -46.

$$x=\frac{-\left(-11\right)±\sqrt{75}}{2}$$

Take the square root of 75.

$$x=\frac{-\left(-11\right)±5\sqrt{3}}{2}$$

The opposite of -11 is 11.

$$x=\frac{11±5\sqrt{3}}{2}$$

Now solve the equation x=\frac{11±5\sqrt{3}}{2} when ± is plus. Add 11 to 5\sqrt{3}.

$$x=\frac{5\sqrt{3}+11}{2}$$

Now solve the equation x=\frac{11±5\sqrt{3}}{2} when ± is minus. Subtract 5\sqrt{3} from 11.

$$x=\frac{11-5\sqrt{3}}{2}$$

The equation is now solved.

$$x=\frac{5\sqrt{3}+11}{2} x=\frac{11-5\sqrt{3}}{2}$$

Steps for Completing the Square

Reduce the fraction \frac{150}{4} to lowest terms by extracting and canceling out 2.

$$\frac{75}{2}=49+x^{2}-11x$$

Swap sides so that all variable terms are on the left hand side.

$$49+x^{2}-11x=\frac{75}{2}$$

Subtract 49 from both sides.

$$x^{2}-11x=\frac{75}{2}-49$$

Subtract 49 from \frac{75}{2} to get -\frac{23}{2}.

$$x^{2}-11x=-\frac{23}{2}$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-\frac{23}{2}+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-\frac{23}{2}+\frac{121}{4}$$

Add -\frac{23}{2} to \frac{121}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-11x+\frac{121}{4}=\frac{75}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{75}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{75}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{5\sqrt{3}}{2} x-\frac{11}{2}=-\frac{5\sqrt{3}}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=\frac{5\sqrt{3}+11}{2} x=\frac{11-5\sqrt{3}}{2}$$