Solve 4x^2+4=98x

$$4 { x }^{ 2 } +4 = 98x$$

Solve for x

$$x = \frac{3 \sqrt{265} + 49}{4} \approx 24.459115447$$

$$x=\frac{49-3\sqrt{265}}{4}\approx 0.040884553$$

Steps Using the Quadratic Formula

Subtract 98x from both sides.

$$4x^{2}+4-98x=0$$

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.

$$4x^{2}-98x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -98 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-98\right)±\sqrt{\left(-98\right)^{2}-4\times 4\times 4}}{2\times 4}$$

Square -98.

$$x=\frac{-\left(-98\right)±\sqrt{9604-4\times 4\times 4}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-98\right)±\sqrt{9604-16\times 4}}{2\times 4}$$

Multiply -16 times 4.

$$x=\frac{-\left(-98\right)±\sqrt{9604-64}}{2\times 4}$$

Add 9604 to -64.

$$x=\frac{-\left(-98\right)±\sqrt{9540}}{2\times 4}$$

Take the square root of 9540.

$$x=\frac{-\left(-98\right)±6\sqrt{265}}{2\times 4}$$

The opposite of -98 is 98.

$$x=\frac{98±6\sqrt{265}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{98±6\sqrt{265}}{8}$$

Now solve the equation x=\frac{98±6\sqrt{265}}{8} when ± is plus. Add 98 to 6\sqrt{265}.

$$x=\frac{6\sqrt{265}+98}{8}$$

Divide 98+6\sqrt{265} by 8.

$$x=\frac{3\sqrt{265}+49}{4}$$

Now solve the equation x=\frac{98±6\sqrt{265}}{8} when ± is minus. Subtract 6\sqrt{265} from 98.

$$x=\frac{98-6\sqrt{265}}{8}$$

Divide 98-6\sqrt{265} by 8.

$$x=\frac{49-3\sqrt{265}}{4}$$

The equation is now solved.

$$x=\frac{3\sqrt{265}+49}{4} x=\frac{49-3\sqrt{265}}{4}$$

Steps for Completing the Square

Subtract 98x from both sides.

$$4x^{2}+4-98x=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-98x=-4$$

Divide both sides by 4.

$$\frac{4x^{2}-98x}{4}=-\frac{4}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{98}{4}\right)x=-\frac{4}{4}$$

Reduce the fraction \frac{-98}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{49}{2}x=-\frac{4}{4}$$

Divide -4 by 4.

$$x^{2}-\frac{49}{2}x=-1$$

Divide -\frac{49}{2}, the coefficient of the x term, by 2 to get -\frac{49}{4}. Then add the square of -\frac{49}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{49}{2}x+\left(-\frac{49}{4}\right)^{2}=-1+\left(-\frac{49}{4}\right)^{2}$$

Square -\frac{49}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{49}{2}x+\frac{2401}{16}=-1+\frac{2401}{16}$$

Add -1 to \frac{2401}{16}.

$$x^{2}-\frac{49}{2}x+\frac{2401}{16}=\frac{2385}{16}$$

Factor x^{2}-\frac{49}{2}x+\frac{2401}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{49}{4}\right)^{2}=\frac{2385}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{49}{4}\right)^{2}}=\sqrt{\frac{2385}{16}}$$

Simplify.

$$x-\frac{49}{4}=\frac{3\sqrt{265}}{4} x-\frac{49}{4}=-\frac{3\sqrt{265}}{4}$$

Add \frac{49}{4} to both sides of the equation.

$$x=\frac{3\sqrt{265}+49}{4} x=\frac{49-3\sqrt{265}}{4}$$