Solve 4x^2-10*36x+5=0

$$4 x ^ { 2 } - 10 \cdot 36 x + 5 = 0$$

Solve for x

$$x=\frac{\sqrt{8095}}{2}+45\approx 89.986108967$$

$$x=-\frac{\sqrt{8095}}{2}+45\approx 0.013891033$$

Steps Using the Quadratic Formula

Multiply 10 and 36 to get 360.

$$4x^{2}-360x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -360 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-360\right)±\sqrt{\left(-360\right)^{2}-4\times 4\times 5}}{2\times 4}$$

Square -360.

$$x=\frac{-\left(-360\right)±\sqrt{129600-4\times 4\times 5}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-360\right)±\sqrt{129600-16\times 5}}{2\times 4}$$

Multiply -16 times 5.

$$x=\frac{-\left(-360\right)±\sqrt{129600-80}}{2\times 4}$$

Add 129600 to -80.

$$x=\frac{-\left(-360\right)±\sqrt{129520}}{2\times 4}$$

Take the square root of 129520.

$$x=\frac{-\left(-360\right)±4\sqrt{8095}}{2\times 4}$$

The opposite of -360 is 360.

$$x=\frac{360±4\sqrt{8095}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{360±4\sqrt{8095}}{8}$$

Now solve the equation x=\frac{360±4\sqrt{8095}}{8} when ± is plus. Add 360 to 4\sqrt{8095}.

$$x=\frac{4\sqrt{8095}+360}{8}$$

Divide 360+4\sqrt{8095} by 8.

$$x=\frac{\sqrt{8095}}{2}+45$$

Now solve the equation x=\frac{360±4\sqrt{8095}}{8} when ± is minus. Subtract 4\sqrt{8095} from 360.

$$x=\frac{360-4\sqrt{8095}}{8}$$

Divide 360-4\sqrt{8095} by 8.

$$x=-\frac{\sqrt{8095}}{2}+45$$

The equation is now solved.

$$x=\frac{\sqrt{8095}}{2}+45 x=-\frac{\sqrt{8095}}{2}+45$$

Steps for Completing the Square

Multiply 10 and 36 to get 360.

$$4x^{2}-360x+5=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-360x=-5$$

Divide both sides by 4.

$$\frac{4x^{2}-360x}{4}=-\frac{5}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{360}{4}\right)x=-\frac{5}{4}$$

Divide -360 by 4.

$$x^{2}-90x=-\frac{5}{4}$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-\frac{5}{4}+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-\frac{5}{4}+2025$$

Add -\frac{5}{4} to 2025.

$$x^{2}-90x+2025=\frac{8095}{4}$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=\frac{8095}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{\frac{8095}{4}}$$

Simplify.

$$x-45=\frac{\sqrt{8095}}{2} x-45=-\frac{\sqrt{8095}}{2}$$

Add 45 to both sides of the equation.

$$x=\frac{\sqrt{8095}}{2}+45 x=-\frac{\sqrt{8095}}{2}+45$$