Solve 25x^2+97=100x

$$25 { x }^{ 2 } +97=100x$$

Solve for x

$$x=\frac{\sqrt{3}}{5}+2\approx 2.346410162$$

$$x=-\frac{\sqrt{3}}{5}+2\approx 1.653589838$$

Steps Using the Quadratic Formula

Subtract 100x from both sides.

$$25x^{2}+97-100x=0$$

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.

$$25x^{2}-100x+97=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -100 for b, and 97 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 25\times 97}}{2\times 25}$$

Square -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\times 25\times 97}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-100\right)±\sqrt{10000-100\times 97}}{2\times 25}$$

Multiply -100 times 97.

$$x=\frac{-\left(-100\right)±\sqrt{10000-9700}}{2\times 25}$$

Add 10000 to -9700.

$$x=\frac{-\left(-100\right)±\sqrt{300}}{2\times 25}$$

Take the square root of 300.

$$x=\frac{-\left(-100\right)±10\sqrt{3}}{2\times 25}$$

The opposite of -100 is 100.

$$x=\frac{100±10\sqrt{3}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{100±10\sqrt{3}}{50}$$

Now solve the equation x=\frac{100±10\sqrt{3}}{50} when ± is plus. Add 100 to 10\sqrt{3}.

$$x=\frac{10\sqrt{3}+100}{50}$$

Divide 100+10\sqrt{3} by 50.

$$x=\frac{\sqrt{3}}{5}+2$$

Now solve the equation x=\frac{100±10\sqrt{3}}{50} when ± is minus. Subtract 10\sqrt{3} from 100.

$$x=\frac{100-10\sqrt{3}}{50}$$

Divide 100-10\sqrt{3} by 50.

$$x=-\frac{\sqrt{3}}{5}+2$$

The equation is now solved.

$$x=\frac{\sqrt{3}}{5}+2 x=-\frac{\sqrt{3}}{5}+2$$

Steps for Completing the Square

Subtract 100x from both sides.

$$25x^{2}+97-100x=0$$

Subtract 97 from both sides. Anything subtracted from zero gives its negation.

$$25x^{2}-100x=-97$$

Divide both sides by 25.

$$\frac{25x^{2}-100x}{25}=-\frac{97}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\left(-\frac{100}{25}\right)x=-\frac{97}{25}$$

Divide -100 by 25.

$$x^{2}-4x=-\frac{97}{25}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{97}{25}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-\frac{97}{25}+4$$

Add -\frac{97}{25} to 4.

$$x^{2}-4x+4=\frac{3}{25}$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=\frac{3}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{3}{25}}$$

Simplify.

$$x-2=\frac{\sqrt{3}}{5} x-2=-\frac{\sqrt{3}}{5}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{3}}{5}+2 x=-\frac{\sqrt{3}}{5}+2$$