Solve x^2+49-74x=0

$$x ^ { 2 } + 49 - 74 x = 0$$

Solve for x

$$x=2\sqrt{330}+37\approx 73.331804249$$

$$x=37-2\sqrt{330}\approx 0.668195751$$

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.

$$x^{2}-74x+49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -74 for b, and 49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-74\right)±\sqrt{\left(-74\right)^{2}-4\times 49}}{2}$$

Square -74.

$$x=\frac{-\left(-74\right)±\sqrt{5476-4\times 49}}{2}$$

Multiply -4 times 49.

$$x=\frac{-\left(-74\right)±\sqrt{5476-196}}{2}$$

Add 5476 to -196.

$$x=\frac{-\left(-74\right)±\sqrt{5280}}{2}$$

Take the square root of 5280.

$$x=\frac{-\left(-74\right)±4\sqrt{330}}{2}$$

The opposite of -74 is 74.

$$x=\frac{74±4\sqrt{330}}{2}$$

Now solve the equation x=\frac{74±4\sqrt{330}}{2} when ± is plus. Add 74 to 4\sqrt{330}.

$$x=\frac{4\sqrt{330}+74}{2}$$

Divide 74+4\sqrt{330} by 2.

$$x=2\sqrt{330}+37$$

Now solve the equation x=\frac{74±4\sqrt{330}}{2} when ± is minus. Subtract 4\sqrt{330} from 74.

$$x=\frac{74-4\sqrt{330}}{2}$$

Divide 74-4\sqrt{330} by 2.

$$x=37-2\sqrt{330}$$

The equation is now solved.

$$x=2\sqrt{330}+37 x=37-2\sqrt{330}$$

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.

$$x^{2}-74x+49=0$$

Subtract 49 from both sides of the equation.

$$x^{2}-74x+49-49=-49$$

Subtracting 49 from itself leaves 0.

$$x^{2}-74x=-49$$

Divide -74, the coefficient of the x term, by 2 to get -37. Then add the square of -37 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-74x+\left(-37\right)^{2}=-49+\left(-37\right)^{2}$$

Square -37.

$$x^{2}-74x+1369=-49+1369$$

Add -49 to 1369.

$$x^{2}-74x+1369=1320$$

Factor x^{2}-74x+1369. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-37\right)^{2}=1320$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-37\right)^{2}}=\sqrt{1320}$$

Simplify.

$$x-37=2\sqrt{330} x-37=-2\sqrt{330}$$

Add 37 to both sides of the equation.

$$x=2\sqrt{330}+37 x=37-2\sqrt{330}$$