Solve x^2-20=5.5x

$${ x }^{ 2 } -20 = 5.5x$$

Solve for x

$$x=8$$

$$x=-2.5$$

Steps Using the Quadratic Formula

Subtract 5.5x from both sides.

$$x^{2}-20-5.5x=0$$

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.

$$x^{2}-5.5x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5.5 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5.5\right)±\sqrt{\left(-5.5\right)^{2}-4\left(-20\right)}}{2}$$

Square -5.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-5.5\right)±\sqrt{30.25-4\left(-20\right)}}{2}$$

Multiply -4 times -20.

$$x=\frac{-\left(-5.5\right)±\sqrt{30.25+80}}{2}$$

Add 30.25 to 80.

$$x=\frac{-\left(-5.5\right)±\sqrt{110.25}}{2}$$

Take the square root of 110.25.

$$x=\frac{-\left(-5.5\right)±\frac{21}{2}}{2}$$

The opposite of -5.5 is 5.5.

$$x=\frac{5.5±\frac{21}{2}}{2}$$

Now solve the equation x=\frac{5.5±\frac{21}{2}}{2} when ± is plus. Add 5.5 to \frac{21}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{5.5±\frac{21}{2}}{2} when ± is minus. Subtract \frac{21}{2} from 5.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{5}{2}$$

The equation is now solved.

$$x=8 x=-\frac{5}{2}$$

Steps for Completing the Square

Subtract 5.5x from both sides.

$$x^{2}-20-5.5x=0$$

Add 20 to both sides. Anything plus zero gives itself.

$$x^{2}-5.5x=20$$

Divide -5.5, the coefficient of the x term, by 2 to get -2.75. Then add the square of -2.75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5.5x+\left(-2.75\right)^{2}=20+\left(-2.75\right)^{2}$$

Square -2.75 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5.5x+7.5625=20+7.5625$$

Add 20 to 7.5625.

$$x^{2}-5.5x+7.5625=27.5625$$

Factor x^{2}-5.5x+7.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2.75\right)^{2}=27.5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2.75\right)^{2}}=\sqrt{27.5625}$$

Simplify.

$$x-2.75=\frac{21}{4} x-2.75=-\frac{21}{4}$$

Add 2.75 to both sides of the equation.

$$x=8 x=-\frac{5}{2}$$