Solve x^2-5.16x-20.7=4.06

$$x ^ { 2 } - 5.16 x - 20.7 = 4.06$$

Solve for x

$$x = \frac{\sqrt{78541} + 129}{50} \approx 8.185033452$$

$$x=\frac{129-\sqrt{78541}}{50}\approx -3.025033452$$

Steps Using the Quadratic Formula

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.

$$x^{2}-5.16x-20.7=4.06$$

Subtract 4.06 from both sides of the equation.

$$x^{2}-5.16x-20.7-4.06=4.06-4.06$$

Subtracting 4.06 from itself leaves 0.

$$x^{2}-5.16x-20.7-4.06=0$$

Subtract 4.06 from -20.7 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5.16x-24.76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5.16 for b, and -24.76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5.16\right)±\sqrt{\left(-5.16\right)^{2}-4\left(-24.76\right)}}{2}$$

Square -5.16 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-5.16\right)±\sqrt{26.6256-4\left(-24.76\right)}}{2}$$

Multiply -4 times -24.76.

$$x=\frac{-\left(-5.16\right)±\sqrt{26.6256+99.04}}{2}$$

Add 26.6256 to 99.04 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-5.16\right)±\sqrt{125.6656}}{2}$$

Take the square root of 125.6656.

$$x=\frac{-\left(-5.16\right)±\frac{\sqrt{78541}}{25}}{2}$$

The opposite of -5.16 is 5.16.

$$x=\frac{5.16±\frac{\sqrt{78541}}{25}}{2}$$

Now solve the equation x=\frac{5.16±\frac{\sqrt{78541}}{25}}{2} when ± is plus. Add 5.16 to \frac{\sqrt{78541}}{25}.

$$x=\frac{\sqrt{78541}+129}{2\times 25}$$

Divide \frac{129+\sqrt{78541}}{25} by 2.

$$x=\frac{\sqrt{78541}+129}{50}$$

Now solve the equation x=\frac{5.16±\frac{\sqrt{78541}}{25}}{2} when ± is minus. Subtract \frac{\sqrt{78541}}{25} from 5.16.

$$x=\frac{129-\sqrt{78541}}{2\times 25}$$

Divide \frac{129-\sqrt{78541}}{25} by 2.

$$x=\frac{129-\sqrt{78541}}{50}$$

The equation is now solved.

$$x=\frac{\sqrt{78541}+129}{50} x=\frac{129-\sqrt{78541}}{50}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-5.16x-20.7=4.06$$

Add 20.7 to both sides of the equation.

$$x^{2}-5.16x-20.7-\left(-20.7\right)=4.06-\left(-20.7\right)$$

Subtracting -20.7 from itself leaves 0.

$$x^{2}-5.16x=4.06-\left(-20.7\right)$$

Subtract -20.7 from 4.06 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5.16x=24.76$$

Divide -5.16, the coefficient of the x term, by 2 to get -2.58. Then add the square of -2.58 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5.16x+\left(-2.58\right)^{2}=24.76+\left(-2.58\right)^{2}$$

Square -2.58 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5.16x+6.6564=24.76+6.6564$$

Add 24.76 to 6.6564 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5.16x+6.6564=31.4164$$

Factor x^{2}-5.16x+6.6564. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2.58\right)^{2}=31.4164$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2.58\right)^{2}}=\sqrt{31.4164}$$

Simplify.

$$x-2.58=\frac{\sqrt{78541}}{50} x-2.58=-\frac{\sqrt{78541}}{50}$$

Add 2.58 to both sides of the equation.

$$x=\frac{\sqrt{78541}+129}{50} x=\frac{129-\sqrt{78541}}{50}$$