Solve 4x^2-388x+5184=0

$$4 { x }^{ 2 } -388x+5184=0$$

Solve for x

$$x=16$$

$$x=81$$

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.

$$4x^{2}-388x+5184=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -388 for b, and 5184 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-388\right)±\sqrt{\left(-388\right)^{2}-4\times 4\times 5184}}{2\times 4}$$

Square -388.

$$x=\frac{-\left(-388\right)±\sqrt{150544-4\times 4\times 5184}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-388\right)±\sqrt{150544-16\times 5184}}{2\times 4}$$

Multiply -16 times 5184.

$$x=\frac{-\left(-388\right)±\sqrt{150544-82944}}{2\times 4}$$

Add 150544 to -82944.

$$x=\frac{-\left(-388\right)±\sqrt{67600}}{2\times 4}$$

Take the square root of 67600.

$$x=\frac{-\left(-388\right)±260}{2\times 4}$$

The opposite of -388 is 388.

$$x=\frac{388±260}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{388±260}{8}$$

Now solve the equation x=\frac{388±260}{8} when ± is plus. Add 388 to 260.

$$x=\frac{648}{8}$$

Divide 648 by 8.

$$x=81$$

Now solve the equation x=\frac{388±260}{8} when ± is minus. Subtract 260 from 388.

$$x=\frac{128}{8}$$

Divide 128 by 8.

$$x=16$$

The equation is now solved.

$$x=81 x=16$$

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.

$$4x^{2}-388x+5184=0$$

Subtract 5184 from both sides of the equation.

$$4x^{2}-388x+5184-5184=-5184$$

Subtracting 5184 from itself leaves 0.

$$4x^{2}-388x=-5184$$

Divide both sides by 4.

$$\frac{4x^{2}-388x}{4}=-\frac{5184}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{388}{4}\right)x=-\frac{5184}{4}$$

Divide -388 by 4.

$$x^{2}-97x=-\frac{5184}{4}$$

Divide -5184 by 4.

$$x^{2}-97x=-1296$$

Divide -97, the coefficient of the x term, by 2 to get -\frac{97}{2}. Then add the square of -\frac{97}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-97x+\left(-\frac{97}{2}\right)^{2}=-1296+\left(-\frac{97}{2}\right)^{2}$$

Square -\frac{97}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-97x+\frac{9409}{4}=-1296+\frac{9409}{4}$$

Add -1296 to \frac{9409}{4}.

$$x^{2}-97x+\frac{9409}{4}=\frac{4225}{4}$$

Factor x^{2}-97x+\frac{9409}{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}.

$$\left(x-\frac{97}{2}\right)^{2}=\frac{4225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{97}{2}\right)^{2}}=\sqrt{\frac{4225}{4}}$$

Simplify.

$$x-\frac{97}{2}=\frac{65}{2} x-\frac{97}{2}=-\frac{65}{2}$$

Add \frac{97}{2} to both sides of the equation.

$$x=81 x=16$$