Solve .35x(76.5-.85x)=40.5

$$.35x(76.5-.85x)=40.5$$

Solve for x

$$x=\frac{45\sqrt{13209}}{119}+45\approx 88.461080822$$

$$x=-\frac{45\sqrt{13209}}{119}+45\approx 1.538919178$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.35x by 76.5-0.85x.

$$26.775x-0.2975x^{2}=40.5$$

Subtract 40.5 from both sides.

$$26.775x-0.2975x^{2}-40.5=0$$

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.

$$-0.2975x^{2}+26.775x-40.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.2975 for a, 26.775 for b, and -40.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-26.775±\sqrt{26.775^{2}-4\left(-0.2975\right)\left(-40.5\right)}}{2\left(-0.2975\right)}$$

Square 26.775 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-26.775±\sqrt{716.900625-4\left(-0.2975\right)\left(-40.5\right)}}{2\left(-0.2975\right)}$$

Multiply -4 times -0.2975.

$$x=\frac{-26.775±\sqrt{716.900625+1.19\left(-40.5\right)}}{2\left(-0.2975\right)}$$

Multiply 1.19 times -40.5 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-26.775±\sqrt{716.900625-48.195}}{2\left(-0.2975\right)}$$

Add 716.900625 to -48.195 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-26.775±\sqrt{668.705625}}{2\left(-0.2975\right)}$$

Take the square root of 668.705625.

$$x=\frac{-26.775±\frac{9\sqrt{13209}}{40}}{2\left(-0.2975\right)}$$

Multiply 2 times -0.2975.

$$x=\frac{-26.775±\frac{9\sqrt{13209}}{40}}{-0.595}$$

Now solve the equation x=\frac{-26.775±\frac{9\sqrt{13209}}{40}}{-0.595} when ± is plus. Add -26.775 to \frac{9\sqrt{13209}}{40}.

$$x=\frac{9\sqrt{13209}-1071}{-0.595\times 40}$$

Divide \frac{-1071+9\sqrt{13209}}{40} by -0.595 by multiplying \frac{-1071+9\sqrt{13209}}{40} by the reciprocal of -0.595.

$$x=-\frac{45\sqrt{13209}}{119}+45$$

Now solve the equation x=\frac{-26.775±\frac{9\sqrt{13209}}{40}}{-0.595} when ± is minus. Subtract \frac{9\sqrt{13209}}{40} from -26.775.

$$x=\frac{-9\sqrt{13209}-1071}{-0.595\times 40}$$

Divide \frac{-1071-9\sqrt{13209}}{40} by -0.595 by multiplying \frac{-1071-9\sqrt{13209}}{40} by the reciprocal of -0.595.

$$x=\frac{45\sqrt{13209}}{119}+45$$

The equation is now solved.

$$x=-\frac{45\sqrt{13209}}{119}+45 x=\frac{45\sqrt{13209}}{119}+45$$

Steps for Completing the Square

Use the distributive property to multiply 0.35x by 76.5-0.85x.

$$26.775x-0.2975x^{2}=40.5$$

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.

$$-0.2975x^{2}+26.775x=40.5$$

Divide both sides of the equation by -0.2975, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.2975x^{2}+26.775x}{-0.2975}=\frac{40.5}{-0.2975}$$

Dividing by -0.2975 undoes the multiplication by -0.2975.

$$x^{2}+\frac{26.775}{-0.2975}x=\frac{40.5}{-0.2975}$$

Divide 26.775 by -0.2975 by multiplying 26.775 by the reciprocal of -0.2975.

$$x^{2}-90x=\frac{40.5}{-0.2975}$$

Divide 40.5 by -0.2975 by multiplying 40.5 by the reciprocal of -0.2975.

$$x^{2}-90x=-\frac{16200}{119}$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-\frac{16200}{119}+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-\frac{16200}{119}+2025$$

Add -\frac{16200}{119} to 2025.

$$x^{2}-90x+2025=\frac{224775}{119}$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=\frac{224775}{119}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{\frac{224775}{119}}$$

Simplify.

$$x-45=\frac{45\sqrt{13209}}{119} x-45=-\frac{45\sqrt{13209}}{119}$$

Add 45 to both sides of the equation.

$$x=\frac{45\sqrt{13209}}{119}+45 x=-\frac{45\sqrt{13209}}{119}+45$$