Solve 2.25x^2-90x+324=0

$$2.25 x ^ { 2 } - 90 x + 324 = 0$$

Solve for x

$$x=4$$

$$x=36$$

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.

$$2.25x^{2}-90x+324=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2.25 for a, -90 for b, and 324 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 2.25\times 324}}{2\times 2.25}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 2.25\times 324}}{2\times 2.25}$$

Multiply -4 times 2.25.

$$x=\frac{-\left(-90\right)±\sqrt{8100-9\times 324}}{2\times 2.25}$$

Multiply -9 times 324.

$$x=\frac{-\left(-90\right)±\sqrt{8100-2916}}{2\times 2.25}$$

Add 8100 to -2916.

$$x=\frac{-\left(-90\right)±\sqrt{5184}}{2\times 2.25}$$

Take the square root of 5184.

$$x=\frac{-\left(-90\right)±72}{2\times 2.25}$$

The opposite of -90 is 90.

$$x=\frac{90±72}{2\times 2.25}$$

Multiply 2 times 2.25.

$$x=\frac{90±72}{4.5}$$

Now solve the equation x=\frac{90±72}{4.5} when ± is plus. Add 90 to 72.

$$x=\frac{162}{4.5}$$

Divide 162 by 4.5 by multiplying 162 by the reciprocal of 4.5.

$$x=36$$

Now solve the equation x=\frac{90±72}{4.5} when ± is minus. Subtract 72 from 90.

$$x=\frac{18}{4.5}$$

Divide 18 by 4.5 by multiplying 18 by the reciprocal of 4.5.

$$x=4$$

The equation is now solved.

$$x=36 x=4$$

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.

$$2.25x^{2}-90x+324=0$$

Subtract 324 from both sides of the equation.

$$2.25x^{2}-90x+324-324=-324$$

Subtracting 324 from itself leaves 0.

$$2.25x^{2}-90x=-324$$

Divide both sides of the equation by 2.25, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{2.25x^{2}-90x}{2.25}=-\frac{324}{2.25}$$

Dividing by 2.25 undoes the multiplication by 2.25.

$$x^{2}+\left(-\frac{90}{2.25}\right)x=-\frac{324}{2.25}$$

Divide -90 by 2.25 by multiplying -90 by the reciprocal of 2.25.

$$x^{2}-40x=-\frac{324}{2.25}$$

Divide -324 by 2.25 by multiplying -324 by the reciprocal of 2.25.

$$x^{2}-40x=-144$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-144+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-144+400$$

Add -144 to 400.

$$x^{2}-40x+400=256$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=256$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{256}$$

Simplify.

$$x-20=16 x-20=-16$$

Add 20 to both sides of the equation.

$$x=36 x=4$$