Solve 1.25x^2-78*5x+361.25=0

$$1.25 x ^ { 2 } - 78 \cdot 5 x + 361.25 = 0$$

Solve for x

$$x=\sqrt{24047}+156\approx 311.070951503$$

$$x=156-\sqrt{24047}\approx 0.929048497$$

Steps Using the Quadratic Formula

Multiply 78 and 5 to get 390.

$$1.25x^{2}-390x+361.25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.25 for a, -390 for b, and 361.25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-390\right)±\sqrt{\left(-390\right)^{2}-4\times 1.25\times 361.25}}{2\times 1.25}$$

Square -390.

$$x=\frac{-\left(-390\right)±\sqrt{152100-4\times 1.25\times 361.25}}{2\times 1.25}$$

Multiply -4 times 1.25.

$$x=\frac{-\left(-390\right)±\sqrt{152100-5\times 361.25}}{2\times 1.25}$$

Multiply -5 times 361.25.

$$x=\frac{-\left(-390\right)±\sqrt{152100-1806.25}}{2\times 1.25}$$

Add 152100 to -1806.25.

$$x=\frac{-\left(-390\right)±\sqrt{150293.75}}{2\times 1.25}$$

Take the square root of 150293.75.

$$x=\frac{-\left(-390\right)±\frac{5\sqrt{24047}}{2}}{2\times 1.25}$$

The opposite of -390 is 390.

$$x=\frac{390±\frac{5\sqrt{24047}}{2}}{2\times 1.25}$$

Multiply 2 times 1.25.

$$x=\frac{390±\frac{5\sqrt{24047}}{2}}{2.5}$$

Now solve the equation x=\frac{390±\frac{5\sqrt{24047}}{2}}{2.5} when ± is plus. Add 390 to \frac{5\sqrt{24047}}{2}.

$$x=\frac{\frac{5\sqrt{24047}}{2}+390}{2.5}$$

Divide 390+\frac{5\sqrt{24047}}{2} by 2.5 by multiplying 390+\frac{5\sqrt{24047}}{2} by the reciprocal of 2.5.

$$x=\sqrt{24047}+156$$

Now solve the equation x=\frac{390±\frac{5\sqrt{24047}}{2}}{2.5} when ± is minus. Subtract \frac{5\sqrt{24047}}{2} from 390.

$$x=\frac{-\frac{5\sqrt{24047}}{2}+390}{2.5}$$

Divide 390-\frac{5\sqrt{24047}}{2} by 2.5 by multiplying 390-\frac{5\sqrt{24047}}{2} by the reciprocal of 2.5.

$$x=156-\sqrt{24047}$$

The equation is now solved.

$$x=\sqrt{24047}+156 x=156-\sqrt{24047}$$

Steps for Completing the Square

Multiply 78 and 5 to get 390.

$$1.25x^{2}-390x+361.25=0$$

Subtract 361.25 from both sides. Anything subtracted from zero gives its negation.

$$1.25x^{2}-390x=-361.25$$

Divide both sides of the equation by 1.25, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.25x^{2}-390x}{1.25}=-\frac{361.25}{1.25}$$

Dividing by 1.25 undoes the multiplication by 1.25.

$$x^{2}+\left(-\frac{390}{1.25}\right)x=-\frac{361.25}{1.25}$$

Divide -390 by 1.25 by multiplying -390 by the reciprocal of 1.25.

$$x^{2}-312x=-\frac{361.25}{1.25}$$

Divide -361.25 by 1.25 by multiplying -361.25 by the reciprocal of 1.25.

$$x^{2}-312x=-289$$

Divide -312, the coefficient of the x term, by 2 to get -156. Then add the square of -156 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-312x+\left(-156\right)^{2}=-289+\left(-156\right)^{2}$$

Square -156.

$$x^{2}-312x+24336=-289+24336$$

Add -289 to 24336.

$$x^{2}-312x+24336=24047$$

Factor x^{2}-312x+24336. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-156\right)^{2}=24047$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-156\right)^{2}}=\sqrt{24047}$$

Simplify.

$$x-156=\sqrt{24047} x-156=-\sqrt{24047}$$

Add 156 to both sides of the equation.

$$x=\sqrt{24047}+156 x=156-\sqrt{24047}$$