Solve 225-540x+324x^2

$$225 - 540 x + 324 x ^ { 2 }$$

Factor

$$9\left(6x-5\right)^{2}$$

Solution Steps

Factor out 9.

$$9\left(25-60x+36x^{2}\right)$$

Consider 25-60x+36x^{2}. Use the perfect square formula, a^{2}-2ab+b^{2}=\left(a-b\right)^{2}, where a=6x and b=5.

$$\left(6x-5\right)^{2}$$

Rewrite the complete factored expression.

$$9\left(6x-5\right)^{2}$$

Steps Using Square Of Binomial

This trinomial has the form of a trinomial square, perhaps multiplied by a common factor. Trinomial squares can be factored by finding the square roots of the leading and trailing terms.

$$factor(324x^{2}-540x+225)$$

Find the greatest common factor of the coefficients.

$$gcf(324,-540,225)=9$$

Factor out 9.

$$9\left(36x^{2}-60x+25\right)$$

Find the square root of the leading term, 36x^{2}.

$$\sqrt{36x^{2}}=6x$$

Find the square root of the trailing term, 25.

$$\sqrt{25}=5$$

The trinomial square is the square of the binomial that is the sum or difference of the square roots of the leading and trailing terms, with the sign determined by the sign of the middle term of the trinomial square.

$$9\left(6x-5\right)^{2}$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$324x^{2}-540x+225=0$$

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.

$$x=\frac{-\left(-540\right)±\sqrt{\left(-540\right)^{2}-4\times 324\times 225}}{2\times 324}$$

Square -540.

$$x=\frac{-\left(-540\right)±\sqrt{291600-4\times 324\times 225}}{2\times 324}$$

Multiply -4 times 324.

$$x=\frac{-\left(-540\right)±\sqrt{291600-1296\times 225}}{2\times 324}$$

Multiply -1296 times 225.

$$x=\frac{-\left(-540\right)±\sqrt{291600-291600}}{2\times 324}$$

Add 291600 to -291600.

$$x=\frac{-\left(-540\right)±\sqrt{0}}{2\times 324}$$

Take the square root of 0.

$$x=\frac{-\left(-540\right)±0}{2\times 324}$$

The opposite of -540 is 540.

$$x=\frac{540±0}{2\times 324}$$

Multiply 2 times 324.

$$x=\frac{540±0}{648}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{6} for x_{1} and \frac{5}{6} for x_{2}.

$$324x^{2}-540x+225=324\left(x-\frac{5}{6}\right)\left(x-\frac{5}{6}\right)$$

Subtract \frac{5}{6} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$324x^{2}-540x+225=324\times \frac{6x-5}{6}\left(x-\frac{5}{6}\right)$$

Subtract \frac{5}{6} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$324x^{2}-540x+225=324\times \frac{6x-5}{6}\times \frac{6x-5}{6}$$

Multiply \frac{6x-5}{6} times \frac{6x-5}{6} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$324x^{2}-540x+225=324\times \frac{\left(6x-5\right)\left(6x-5\right)}{6\times 6}$$

Multiply 6 times 6.

$$324x^{2}-540x+225=324\times \frac{\left(6x-5\right)\left(6x-5\right)}{36}$$

Cancel out 36, the greatest common factor in 324 and 36.

$$324x^{2}-540x+225=9\left(6x-5\right)\left(6x-5\right)$$

Evaluate

$$9\left(6x-5\right)^{2}$$